- Email: [email protected]
The effect of thermal boundary conditions on a stability of convective flow in a horizontal layer subjected to the longitudinal temperature gradient and acoustic wave
International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The effect of thermal boundary conditions on a stability of convective flow in a horizontal layer subjected to the longitudinal temperature gradient and acoustic wave T.P. Lyubimova a,b,⇑, R.V. Skuridin a a b
Institute of Continuous Media Mechanics UB RAS, Koroleva Str., 1, 614013 Perm, Russia Perm State University, Bukireva Str., 15, 614990 Perm, Russia
a r t i c l e
i n f o
Article history: Received 19 May 2015 Received in revised form 6 August 2015 Accepted 6 August 2015
a b s t r a c t The effect of thermal boundary conditions on linear stability of stationary convective flow in a horizontal layer subjected to the longitudinal temperature gradient and propagating acoustic wave is studied. Twodimensional shear instability and three-dimensional monotonic and oscillatory instabilities are analyzed. The stability maps for different acoustic Reynolds number values with respect to different perturbation types are obtained. The stabilization under the influence of acoustic wave at low Prandtl numbers is demonstrated. It takes place when two-dimensional shear or three-dimensional oscillatory instability modes are most dangerous. Three-dimensional monotonic mode is destabilized by acoustic influence. Energy analyses for two-dimensional and monotonic three-dimensional modes of instability is carried out. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The problem of thermal buoyancy convection in the presence of horizontal temperature gradient has multiple practical applications and is widely studied. Such flows appear in the number of technological processes and geophysical phenomena, e.g. some types of flows in the ocean, crust and mantle of the Earth, melt flows in the production of crystals. In the latter case, the character of heat and mass transfer in the melt influences considerably the microstructure of the growing crystal (see [1]). That is why understanding of physical mechanisms, determining dynamics of the melt and its stability is important. In many cases suppression of development of primary instability leads to stabilization of the process in general, which is very desirable. The problem of stability of convective flow in the horizontal layer with rigid perfectly conductive boundaries subjected to the longitudinal temperature gradient is widely studied (see [2,3–7]). As analysis showed, there are several mechanisms of instability. What mechanism is most dangerous, depends on the value of Prandtl number. At low Prandtl numbers Pr < 0:14 twodimensional shear instability mechanism is at work (perturbations are vortices, generated on the boundary of opposing directions flows and look as rolls with axes perpendicular to the gradient of ⇑ Corresponding author at: Institute of Continuous Media Mechanics UB RAS, Koroleva Str., 1, 614013 Perm, Russia. E-mail address: [email protected] (T.P. Lyubimova). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.019 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
temperature). In the Prandtl number range 0:14 < Pr < 0:44 the three-dimensional oscillatory perturbations are most dangerous. These perturbations are internal waves generated in the layer of stable stratification due to energy of basic flow. They represent the rolls with axes parallel to the direction of temperature gradient. At Pr > 0:44 the monotonic three-dimensional modes are the most dangerous. These are perturbations having the form of rolls with axes parallel to the temperature gradient, generated in the area of unstable temperature stratification, i.e. the instability mechanism is of the Rayleigh type. The case of adiabatic boundaries was considered in [8]. It was shown that in this case two-dimensional shear instability is most dangerous in a narrower range of Prandtl number values: at Pr < 0:033. As in the case of perfectly conductive boundaries, with the increase of Prandtl number this instability mode is replaced by three-dimensional oscillatory perturbations which remain the most dangerous at 0:033 < Pr < 0:21. Differ from the case of perfectly conductive boundaries there are no domains of unstable temperature stratification in the basic state temperature profile and due to that the monotonic three-dimensional instability mode of the Rayleigh type does not exist. Instead of that, at Prandtl number larger than 0.21 there arises new instability mode which does not exist for perfectly conductive boundaries. This is monotonic three-dimensional instability mode. The critical Rayleigh numbers for this mode is substantially higher than that for two instability modes discussed above, however due to the sharp stabilization of those modes at Pr 0:033 and Pr 0:21 respectively, at
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Pr > 0:21 the three-dimensional monotonic instability mode is most dangerous. In [9,10] convection in a horizontal fluid layer under inclined temperature gradient was studied. Perfectly conductive boundaries were considered. Linear stability with respect to multiple modes for wide range of Pr was investigated. Analytical expression for the limit Pr ! 1 was obtained. The linear stability of convective flow in the horizontal channel of rectangular cross-section with respect to the perturbation modes similar to perturbations in the horizontal layer, depending on the Prandtl number and channel width, was studied numerically in [11]. It is shown that unidirectional (longitudinal) steady flow in a channel exists only for Pr ¼ 0, at any non-zero Prandtl number values the flow is three-dimensional. Stability maps are obtained, structure of critical perturbations is examined and analyzed. The transition to the limit case of infinitely large width channel (layer) is observed, the qualitative modification of the structure and physical nature of perturbations as channel width diminishes is shown. The presence of large variety of instability mechanisms allows application of different supplementary factors (motion of boundaries, stationary and rotating magnetic fields and others) for control of stationary convective flow stability. One of the promising methods is the use of acoustic wave. It is known (see [12]) that acoustic wave can generate stationary flows in viscous fluid. The effect of plane propagating acoustic wave on the convective flows in the horizontal layer with perfectly conductive boundaries was studied in [13]. The linear stability of stationary flow to twodimensional and three-dimensional perturbations was investigated for low and moderate Prandtl number values. Stability maps for two-dimensional and three-dimensional perturbations are obtained. It is found that the acoustic wave of low enough intensity (small enough values of Reynolds number) makes stabilizing effect; weakly non-linear analysis has shown that all considered types of instability are excited through supercritical bifurcation. The effect of ultrasound beam on the stability of convective flow in the horizontal layer in the case when the layer thickness is much larger than acoustic wave length (which means either larger system dimension or higher sound frequency), was studied in [14]. In this case the beam creates volumetric force, generating Eckart streaming in the fluid. The calculations have shown the presence of stabilizing effect at low and moderate Prandtl number values, significant attention was paid to the influence of the beam width and its position in the layer cross-section. From the point of view of application to the horizontal directional solidification the problem of acoustics effect on the stability of convective flow in horizontal layer with adiabatic boundaries is important. This problem was not studied earlier. Its investigation under assumption of comparable acoustic wavelength and characteristic problem dimension is the goal of the present work. To analyze the effect of thermal properties of layer boundaries we carry out calculations for perfectly conductive boundaries and adiabatic boundaries in parallel.
2. Problem formulation The horizontal layer of fluid, limited from above and from below by rigid horizontal plates located at the distance H from each other is under consideration. Coordinate axis z is directed horizontally and located in-between plates at distance H=2. Axis y is directed vertically upwards, axis x is directed horizontally and perpendicularly to the axis z. Temperature, linearly varying on the longitudinal coordinate, is imposed on the plates: T ¼ Az, where A is a constant. Plane acoustic wave, velocity of which is described by w ¼ axk cosðxðt z=cÞÞ, is propagating in the
1047
direction of temperature gradient and axis z (here a is the amplitude, x is the frequency, c is the velocity of the sound, k is the unit vector of the axis z). The oscillation period is assumed to be small compared to characteristic viscous and thermal time scales. This allows to use averaged description. The governing equations for thermo-acoustic convection were obtained by averaging method in [15]. For the acoustic wave under consideration, due to the spatial uniformity of acoustic field, the equations for average components look similar to the conventional equations of thermal buoyancy convection in the Boussinesq approximation and the effect of acoustic wave which generates the average vorticity in boundary layers near rigid walls of the channel, spreading all over the volume of the fluid due to convection and viscosity and leading to formation of acoustic flow is described by means of effective boundary conditions at the outer border of boundary layer. These boundary conditions are translated directly to the rigid surface, since boundary layer thickness is small. Analysis gives the value v z ¼ w ¼ 3a2 x2 =ð4cÞ for the velocity on the walls of the channel [15,16].
Taking quantities H, m=H, H2 =m, AH and qm2 =H2 as scales for length, velocity, time, temperature and pressure respectively, we write down the system of equations for average momentum, energy and continuity in the form [15]:
@ v =@t þ ðv rÞv ¼ rp þ Dv þ GrT c;
ð1Þ
@T=@t þ ðv rÞT ¼ DT=Pr;
ð2Þ
div
v ¼ 0;
ð3Þ
c is the unit vector of the axis y. The boundary condition for average velocity on the layer boundaries (at y = ±1/2) is
v ¼ Re k
ð4Þ
Additionally, supposing closeness of layer at infinity, we impose the condition of no-flow through cross-section:
Z
1=2
1=2
v z dy ¼ 0:
ð5Þ
In the limit case of perfectly conductive boundaries the boundary condition for temperature is
T ¼ z:
ð6Þ
In the opposite limit case of adiabatic boundaries it is replaced by
@T=@y ¼ 0:
ð7Þ
The problem is characterized by following dimensionless parameters: Gr ¼ gbAH4 =m2 is the Grashof number, Pr ¼ m=v is the Prandtl number, Re ¼ 3a2 x2 H=ð4cmÞ is the pulsational Reynolds number. Here m is the kinematic viscosity, q is the fluid density, g is the gravity acceleration, v is the thermal diffusivity. The problem under consideration permits stationary solution, corresponding to plane-parallel flow, where velocity is uniform along z axis and transversal components are absent. In this case system (1)–(3) reduces to
@p0 =@y þ GrT 0 ¼ 0;
ð8Þ
@p0 =@z þ @ 2 v 0z =@y2 ¼ 0;
ð9Þ
v 0z ¼ @ 2 T 0 =@y2 =Pr:
ð10Þ
Solution for longitudinal component of velocity is a superposition of convective and acoustic flows:
v 0z w0 ¼ Grðy3 y=4Þ=6 þ Reð6y2 1=2Þ:
ð11Þ
1048
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Stationary profile of temperature in the case of perfectly conductive boundaries (problem (8)–(10), (4)–(6)) is
there is a single zone of unstable stratification in the upper part of the layer (Fig. 1f).
T 0 ¼ z þ Re Prðy4 =2 y2 =4 þ 1=32Þ
3. Linear stability problem
þ Gr Prðy5 =120 y3 =144 þ 7y=5760Þ;
ð12Þ
To study stability of basic state we represent the temperature, velocity and pressure fields in the form of sums of basic state and small perturbations, introduce the fields in such a form into original equations and linearize obtained equations with respect to small perturbations. The presence of longitudinal temperature gradient prohibits acquisition of transformation, reducing the problem of three-dimensional perturbations to the problem of two-dimensional perturbations, and in general case it is necessary to consider complete formulation. However, as it is demonstrated below, perturbations, which are of one or another limiting case, are the most dangerous in most areas of parameter space. The coefficients in linearized equations for small perturbations do not depend on time and x and z coordinates (the temperature T 0 depend on z, but its z-derivative, which enters into linearized energy equation for perturbations, is equal to 1). This allows us to consider perturbations of all fields in the form f ðyÞ expðkt þ ikx x þ ikz zÞ. It is convenient to denote kx ¼ k cosðaÞ,
and in the case of adiabatic boundaries (problem (8)–(10), (4), (5), (7)) is:
T 0 ¼ z þ Re Prðy4 =2 y2 =4 þ 1=32Þ þ Gr Prðy5 =120 y3 =144 þ y=384Þ
ð13Þ
Stationary profiles of longitudinal velocity component and temperature with z subtracted at Reynolds number equal to 1000 and different values of Prandtl and Grashof numbers for the cases of perfectly conductive and adiabatic boundaries are presented in Fig. 1. Grashof numbers for this figure correspond to the stability thresholds at given Prandtl number values, and Prandtl numbers are selected in order to illustrate typical structures of the solution. In the case of perfectly conductive boundaries purely convective or convection-dominated flow is characterized by the presence of 0 zones of potentially unstable stratification with @T < 0, @@yq > 0 near @y upper and lower boundaries of the layer, while in its central part 0 temperature distribution does not create instability @T > 0, @y @q @y
kz ¼ k sinðaÞ, kx þ kz k , where a is the angle between wave vector and x axis. Introducing notations u, v , w, h, P for amplitudes of perturbations of x-, y-, z-components of velocity, temperature and pressure respectively we obtain the following spectral-amplitude problem (stroke denotes derivation by y): 2
< 0, Fig. 1a). As relative role of acoustic mechanism of flow gen-
eration increases, Gr number grows, the lower area of potential instability diminishes while the upper grows (Fig. 1b). When acoustic flow dominates, only one area of potential instability remains (Fig. 1c). In the case of adiabatic boundaries convective flow does not generate similar zones of potential instability (Fig. 1d). As contribution of acoustic component of flow increases such zone starts to develop in the upper part of the layer (Fig. 1e). Finally, when the flow becomes acoustics-dominated, situation becomes similar to the case of perfectly conductive boundaries, because acoustic component does not depend on the boundary conditions type,
T0
-0.4 -0.2
0
2
2
ku þ ikz w0 u ¼ ikx P k u þ u00 ;
ð14Þ
kv þ ikz w0 v ¼ P0 k
ð15Þ
2
ð16Þ
kh þ ikz w0 h þ v T 00 þ w ¼ ðh00 k hÞ=Pr;
ð17Þ
2
2
T0
0.2
0.4
-0.4 -0.2
0
T0
0.2
0.4
-40
0.4
0.4
0.2
0.2
0.2
y 0
y 0
y 0
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
0 v0z
4000
-1000
(a) -0.8 -0.4
0
0 v0z
1000
-1000
0.4
0.8
-0.4 -0.2
0
0.4
-40
0.4
0.4
0.2
0.2
y 0
y 0
y 0
-0.2
-0.2
-0.2
-0.4 0 v0z
(d)
4000
20
0 v0z
40
1000
T0
0.2
0.2
-0.4
0
(c)
T0
0.4
-4000
-20
(b)
T0
v þ v 00 þ Grh;
kw þ ikz w0 w þ w00 v ¼ ikz P k w þ w00 ;
0.4
-4000
2
-20
0
20
40
-0.4 -1000
0 v0z
(e)
1000
-1000
0 v0z
1000
(f)
Fig. 1. Profiles of longitudinal component of velocity (solid lines) and temperature minus z (dashed lines) for Re ¼ 1000, perfectly conductive boundaries ((a) Pr ¼ 0:005, Gr ¼ 288; 830, (b) Pr ¼ 0:0088, Gr ¼ 69; 740, (c) Pr ¼ 1, Gr ¼ 51), and adiabatic boundaries ((d) Pr ¼ 0:002, Gr ¼ 412; 810, (e) Pr ¼ 0:005, Gr ¼ 70; 000, (f) Pr ¼ 1, Gr ¼ 28:08).
1049
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
ikx u þ v 0 þ ikz w ¼ 0; y ¼ 1=2 :
ð18Þ
u ¼ v ¼ w ¼ 0;
Table 1 Critical Grashof number for different modes with different nodes numbers for Pr ¼ 0:09, Re ¼ 1000, adiabatic boundaries.
ð19Þ
Additionally, in the case of perfectly conductive boundaries we have:
y ¼ 1=2 :
h¼0
ð20Þ
Nodes number
51
75
101
201
2D 3D oscillatory 3D monotonic
119,000 15,040 541.2
115,400 15,070 544.4
114,100 15,070 545.7
113,100 15,080 546.8
and in the case of adiabatic boundaries:
y ¼ 1=2 :
h0 ¼ 0;
To estimate grid convergence, Table 1 is provided. As one can see, 101 nodes seem to be adequate.
ð21Þ
Closeness condition (5) for perturbations satisfies automatically if kx ¼ 0. Otherwise flow closure is ensured by periodicity of perturbations in x-direction. The limit case a ¼ p=2 (kx ¼ 0) corresponds to two-dimensional perturbations in the form of rolls with axes perpendicular to basic flow (Fig. 2a), where as seen from (14) u ¼ 0, the limit case a ¼ 0 (kz ¼ 0) corresponds to three-dimensional perturbations in the form of rolls with axes parallel to basic flow (Fig. 2b). In the intermediate case kx –0, kz –0 there are perturbations of inclined waves type. The problem is not changed at simultaneous replacement of y by y, k by k, P by P and Re by Re. At the same time simultaneous replacement of Gr and Re sign doesn’t lead to the physically new situation. Consequently, it is sufficient to consider only the case of positive values of Grashof and Reynolds numbers. Solution of spectral problem at given values of other parameters (Gr, Pr, Re, k, a) allows to obtain the increment k and amplitudes of perturbations as eigenvalues and eigenvectors of the problem. Search of Grashof number value at which real part of k takes zero value gives stability threshold with respect to a given mode. Minimization of this number as function of k (and a in the case of inclined waves) allows to obtain the map of stability of basic flow. Spectral-amplitude problem was discretized by finite difference method of second order of accuracy on the uniform grid with 101 nodes, which reduced it to generalized algebraic problem of eigenvalues AX = k BX, where the matrix A is complex and has a band structure, the matrix B is diagonal, with part of the diagonal elements equal to unity and the others equal to zero, and X is the vector of the unknown perturbations. Computational package developed in [17] was used for the solution which uses the Newton–Raphson method to iteratively find eigenvalues and eigenvectors. Resulting system of linear algebraic equations is solved by LU-decomposition using standard package of linear algebra Lapack. The first guess for eigenvalues and eigenvectors needed for implementation of Newton–Raphson method was obtained by direct numerical simulation of temporal evolution of initial perturbations based on unsteady linearized equations until the evolution is only determined by the most dangerous mode.
4. Numerical results 4.1. Two-dimensional shear instability (kx ¼ 0) Two-dimensional shear instability is related to the development of perturbations in the form of vortices on the boundary of counter streams (transverse rolls). This mode, which is monotonic at Re ¼ 0 (i.e. its imaginary part of k is equal to zero), becomes oscillatory in the presence of acoustic wave. This mode is stabilized by acoustics; stability maps for perfectly conductive and adiabatic boundaries (Fig. 3, logarithmic scale don’t allow to explicitly show data for Pr ¼ 0, but they nearly coincide with the case Pr ¼ 0:001 at the left edge of charts) are similar to each other qualitatively (at Pr ¼ 0 problems coincide). Quantitatively, in the case of adiabatic boundaries as Prandtl number grows, this mode is stabilized more quickly, the flow becomes absolutely stable relative to this mode at Pr 0:32 for perfectly conductive boundaries and at Pr 0:15 for adiabatic boundaries. Approximately twofold difference in maximal Prandtl number values, at which instability with respect to this mode is observed, can be explained by similar (also approximately two-folded) difference in vertical temperature gradient between counter streams, prohibiting development of this mode (see (12) and (13)). Instability of purely acoustic flow (at Gr ¼ 0) to twodimensional shear instability mode was not found, supposedly it is absolutely stable. Graphs of wavenumbers k and frequencies x as function of Prandtl number are plotted in Figs. 4 and 5. They demonstrate qualitatively similar behavior both for kðPrÞ and xðPrÞ for both types of thermal boundaries conditions, just the Prandtl number values at which sharp increase of k and decrease of x occurs in the course of sharp stabilization of this mode with the increase of Pr differs (Pr 0:32 for perfectly conductive boundaries and Pr 0:15 for adiabatic). Critical wavenumbers are bigger for high Pr in the case of adiabatic boundaries, while qualitatively observed pictures are similar. Wavenumbers almost converge at Pr ¼ 0:2 and Pr ¼ 0:1 for considered boundaries conditions variants respectively, and
y
z
x
(a)
(b)
Fig. 2. Schematics of flow structure in the case of two-dimensional (a) and three-dimensional (b) perturbations.
1050
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Gr
Gr
1x10
5
1x104
0.001
0.01
(a)
0.1
1x10
5
1x10
4
Pr
0.001
Re=0 Re=200
0.01
(b)
Re=400 Re=600
Pr
0.1
Re=800 Re=1000
Fig. 3. Maps of stability with respect to two-dimensional shear instability mode of perturbations in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries.
3.2 k
3.2 k
2.8
2.8
2.4
2.4
2 0.001
0.01
(a)
0.1
Pr
Re=0 Re=200
2 0.001
0.01
(b)
Re=400 Re=600
0.1
Pr
Re=800 Re=1000
Fig. 4. Critical wavenumber k for two-dimensional shear instability mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
ω
0
ω
0
-400
-400
-800
-800
-1200 0.001
0.01
(a)
Re=200
0.1
Pr
-1200 0.001
Re=400 Re=600
0.01
(b)
0.1
Pr
Re=800 Re=1000
Fig. 5. Oscillation frequency x for two-dimensional shear instability mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1051
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
then grow. Charts of critical frequencies are also qualitatively similar, frequencies monotonously grow as Re increases. 4.2. Three-dimensional oscillatory mode with kz ¼ 0 Three-dimensional oscillatory perturbations are traveling waves, propagating in the direction perpendicular to the temperature gradient (longitudinal rolls), in the area of stable stratification. Families of stability curves for different Reynolds number values are presented in Fig. 6. At low Reynolds number values stability maps for both variants of thermal boundary conditions are qualitatively similar, at higher Reynolds numbers in the case of adiabatic boundaries their behavior becomes much more complicated. Quantitative differences are high enough even at Re ¼ 0. Let us discuss the results in more detail. For purely convective flow (Re ¼ 0) at low Prandtl numbers critical Grashof number at first decreases as Pr increases (in logarithmic coordinates this dependence is linear and in the limit Pr ! 0 approaches asymptotes lnðGr C Þ 1:05 lnðPrÞ þ 7:15 for perfectly conductive boundaries and lnðGr C Þ 0:494 lnðPrÞþ 7:416 for adiabatic boundaries), reaches minimum (at Pr ¼ 0:288, Gr C ¼ 12; 150 and Pr ¼ 0:103, Gr C ¼ 7332 respectively), then it starts to grow and tends to vertical asymptotes Pr ¼ 0:456 and Pr ¼ 0:207 respectively. At non-zero Reynolds numbers asymptote is not modified as Re grows, but in the vicinity of the stability curve
Gr
Gr θ
w 1x10
minimum the mode is monotonously stabilized with simultaneous decrease of Pr, at which this minimum is attained. For perfectly conductive boundaries this behavior remains at any Re, but in case of adiabatic boundaries at Re > 450 another branch of the same mode appears, which is even destabilized with Re growth in some range of Prandtl numbers ( 0:04 < Pr < 0:15). In the inset of Fig. 6b the structure of both branches of perturbations for Re ¼ 1000, Pr ¼ 0:045, Gr ¼ 18; 550 is presented, on the left for the branch starting in the area of low Prandtl numbers, on the right for the branch which continues in the area of relatively high Prandtl numbers. As it is seen, depending on the boundary conditions type, the structure of temperature perturbations changes drastically. There are two minima on the neutral curves in some zone of parametric space (Fig. 7), and as Prandtl number varies, one or other of them becomes global. All neutral curves intersect in the vicinity of point Gr 19; 000, k 0:85, but apparently there is nothing particular about it. Critical wavenumbers and frequencies are presented in Figs. 8 and 9. While at high Pr there is some qualitative similarity of behavior in two boundary conditions variants considered, at low Prandtl numbers differences are significant, especially at Pr ¼ 0:01 critical frequencies are approximately order of magnitude higher in the case of perfectly conductive boundaries and are decreasing with Pr growth, while in the case of adiabatic boundaries they are almost constant.
w
w
θ
θ
5
1x10
5
w 1x104
θ
w
0.01
1x10 0.1
(a)
Pr
0.01
Re=0 Re=200
θ
4
Pr
0.1
(b)
Re=400 Re=600
Re=800 Re=1000
Fig. 6. Maps of stability relative to three-dimensional oscillatory mode of perturbations in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries. Profiles of longitudinal component of velocity and temperature perturbations are presented in the insets (w, T, solid and dashed lines are real and imaginary parts) at the parametric space points, indicated by arrows.
Gr 21000 20000 19000 18000 17000 0.4
0.8 Pr=0.035 Pr=0.04
Pr=0.045 Pr=0.05
Fig. 7. A family of neutral curves for adiabatic boundaries, Re ¼ 1000.
k 1.2 Pr=0.055
1052
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
2
2
k
k
1.5
1.5
1
1
0.5
0.5
0 0.01
0.1
(a)
Pr
Re=0 Re=200
0 0.01
Pr
0.1
(b)
Re=400 Re=600
Re=800 Re=1000
Fig. 8. Critical wavenumber k for three-dimensional oscillatory mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
In the case of perfectly conductive boundaries at low Pr wavelength of three-dimensional oscillatory perturbations virtually does not depend on Prandtl number, remaining close to k 2, while frequency monotonously grows as Pr diminishes, similar to critical value of Gr. In the case of adiabatic boundaries wavenumber grows up as Pr grows, reaching some maximum. As Pr approaches the threshold of absolute stabilization of the mode for both variants of boundary conditions wavenumber tends to 0 (and wavelength correspondingly to infinity), while frequency tends to some limit value (x ¼ 151 for conductive boundaries and x ¼ 47:6 for adiabatic boundaries). Similar to critical Grashof numbers, this asymptote weakly depends on Reynolds number, i.e. at low and extremely high Prandtl numbers characteristics of perturbations are determined mostly by convective mechanism of flow generation. At Gr ¼ 0 (purely acoustic flow) the exact solution of the problem for perturbations can be obtained, it turns out that the only solution is trivial solution. So, acoustic flow is absolutely stable with respect to longitudinal rolls type instabilities. This conclusion is correct for both monotonic and oscillatory three-dimensional perturbations. Similar result was obtained numerically in [14] for the acoustic beam case.
4.3. Mode of inclined waves type (kx –0, kz –0) Perturbations of inclined waves type can be described as the result of transformation of three-dimensional oscillatory perturbations with kz ¼ 0 when the requirement of orthogonality of wavevector to temperature gradient is removed. Critical angle a in the domain of Pr values where this mode can be the most dangerous does not exceed the value a ¼ 0:3 ¼ 17 (and usually is smaller), so structures of perturbations with kz ¼ 0, kx –0 and with kz –0, kx –0 are similar. Maps of stability (Fig. 10) are also similar (decreasing of critical Gr, as Pr grows, linear in logarithmic coordinates and reaching the minimum, then grows tending to vertical asymptote), however in the case of adiabatic boundaries there are qualitative differences. Stability threshold with respect to this mode is always lower than the threshold with respect to oscillatory longitudinal rolls. At low values of Prandtl and Reynolds numbers (approximately up to Pr 0:01 and Re 250) mode of inclined waves type is destabilized as intensity of acoustic wave grows (very weakly for perfectly conductive boundaries and slightly stronger for adiabatic boundaries). At high values of any of these parameters mode is stabilized. At perfectly conductive boundaries, limiting value of
400
160
ω
ω 120
300 80 200 40
100 0.01
0.1
(a)
Re=0 Re=200
Pr
0 0.01
Re=400 Re=600
Pr
0.1
(b)
Re=800 Re=1000
Fig. 9. Oscillation frequency x for three-dimensional oscillatory mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1053
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
1x106 Gr
1x106 Gr
1x10
5
1x10
5
1x10
4
1x10
4
0.001
0.01
0.1
Pr
0.001
0.01
(a)
Pr
0.1
(b)
Re=0 Re=200
Re=400 Re=600
Re=800 Re=1000
Fig. 10. Maps of stability with respect to mode of perturbations of inclined waves type in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries.
Prandtl number, at which flow becomes absolutely stable with respect to this mode, is Pr 0:65, about 30% higher than for three-dimensional oscillatory mode with kz ¼ 0. Critical angle of inclination of wavevector here tends to some value a 0:75 ¼ 43 . For adiabatic boundaries instability exists in a wider range of Prandtl number values than for three-dimensional oscillatory mode with kz ¼ 0, stability curve reaches Pr 0:22 (difference from three-dimensional oscillatory mode with kz ¼ 0 is less than 5%), but then declines backwards to lower Pr, approaching the same vertical asymptote Pr ¼ 0:207 as threedimensional oscillatory mode with kz ¼ 0. So, in narrow range of Pr flow is stabilized as Gr grows. Critical wavenumbers and frequencies are plotted in Figs. 11 and 12. In general, the curves are comparable with ones for oscillatory longitudinal rolls. For perfectly conductive boundaries the mode of inclined waves type exists even for Re ¼ 0 if Pr < 0:034 (at higher Prandtl numbers a ¼ 0, mode coincides with three-dimensional oscillatory mode) and is shown in Fig. 8 although merges with curves for the other Re values. For adiabatic boundaries, and Re ¼ 0 these modes coincide for any Pr, splitting occurs only for Re > 0. For adiabatic boundaries at Re > 450 there is additionally specific branch of this mode existing near minimum of stability
2 k
curve, more dangerous than principal branch in some range of Prandtl numbers in the vicinity of Pr ¼ 0:1 and is destabilized as Re grows. Critical wavenumbers and frequencies behavior for the mode under consideration is similar to that for oscillatory longitudinal rolls. Due to qualitative similarity of this mode to three-dimensional oscillatory mode with kz ¼ 0, pure acoustic flow probably is also absolutely stable with respect to it, although it is difficult to strictly demonstrate this statement. 4.4. Three-dimensional monotonic instability mode with kx –0, kz ¼ 0 of Rayleigh origin For Re ¼ 0 there are two three-dimensional monotonic instability modes with kx –0, kz ¼ 0 (longitudinal rolls), which differ from each other in symmetry. Even mode (signs of perturbations of longitudinal component of velocity and temperature are identical near upper and lower boundaries of layer) is more dangerous, and we will restrict ourselves by consideration of it (odd mode lies higher and is stabilized with Re growth). If Re–0, this mode, obviously, does not possess any definite symmetry. For Pr < 0:1 pictures for perfectly conductive and adiabatic boundaries are quite similar,
2 k
1.6
1.5
1.2 1 0.8 0.5
0.4 0 0.001
0.01
0.1
Pr
0 0.001
(a) Re=0 Re=200
0.01
0.1
Pr
(b) Re=400 Re=600
Re=800 Re=1000
Fig. 11. Critical wavenumber k for mode of perturbations of inclined waves type as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1054
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
1000 ω
120 ω
800
100
600
80
400
60
200
40
0 0.001
0.01
20 0.001
Pr
0.1
0.01
(a)
Pr
0.1
(b)
Re=0 Re=200
Re=400 Re=600
Re=800 Re=1000
Fig. 12. Oscillation frequency x for mode of perturbations of inclined waves type as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
for higher Prandtl numbers in the case of adiabatic boundaries stability map significantly complicates, qualitative differences arise. Let us consider at first the case of perfectly conductive boundaries (Fig. 13a). For Re ¼ 0 critical Grashof number monotonously and (in logarithmic coordinates) almost linearly decreases with Prandtl number growth from Gr ¼ 1; 170; 000 for Pr ¼ 0:001 to Gr ¼ 1360 for Pr ¼ 10. Energy of perturbations concentrates mostly near the walls in the area of unstable stratification of temperature (this mode is of Rayleigh-type nature). In the central part of the layer there is a weaker vortex of opposite direction. For non-zero Reynolds numbers and low Pr the mode is weakly destabilized with Re growth. In this case transversal gradient of temperature in the area of unstable stratification in the upper part of the layer grows, while diminishing in the lower part, and correspondingly the energy of perturbations relocates to the upper part of the channel (the upper vortex intensifies and the lower vortex diminishes), while a stagnation zone forms in the lower part. For high Pr it is more correct to speak about destabilization of acoustic flow with Gr growth (flow is mostly generated by acoustic
Gr 1x10
w
θ
5
θ
mechanism on the stability threshold), but despite completely different structure of basic flow, perturbation energy also concentrates in the upper part of the layer in the zone of unstable stratification, although stagnation zone is less pronounced. Stability curves once again are approximately linear in logarithmic coordinates, but lay lower and at a steeper angle and destabilization by acoustic wave is more substantial. In between ranges of ‘low Prandtl numbers’ and ‘high Prandtl numbers’ there is S-shaped transitional section. As Re grows, it translates to lower Pr and becomes steeper. The map of stability to this mode for adiabatic boundaries is presented in Fig. 13b. In the case of adiabatic boundaries area of unstable stratification does not exist due to convective flow. However, as it was shown in [8], at Pr > 0:21 there exists three-dimensional monotonic instability mode with kz ¼ 0, kx –0, which is not related to the existence of the instable temperature stratification. For Re ¼ 0 and low Pr dependence of critical Grashof number on Prandtl number is qualitatively similar to the dependence for perfectly conductive boundaries considered above, with a little bit weaker
Gr
θ
1x10
θ
w
w θ
1x103
w
1x103
θ
θ
w
w
θ
w
w 1
1x10 0.001
w
5
θ
w
θ
1
0.01
0.1
1
1x10 Pr 10 0.001
(a) Re=0 Re=50 Re=100
0.01
0.1
1
Pr 10
(b) Re=167.7 Re=200 Re=400
Re=600 Re=800 Re=1000
Fig. 13. Maps of stability relative to three-dimensional monotonic mode of perturbations in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries. Profiles of longitudinal component of velocity and temperature perturbations are presented in the insets (w is solid and T is dashed line) in the parameter space points, indicated by arrows.
1055
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
stability, but at Pr 0:15 the minimum Gr c ¼ 130; 000 is reached, and later Gr c increases since the absence of area of unstable temperature stratification leads to mode stabilization. Thus, this mode is not of Rayleigh-type nature and exists in the area where deviation of temperature from conductive distribution is maximal, and convective transfer, which creates it, diminishes. Perturbations concentrate near layer boundaries to even stronger degree, while being almost absent in central part of it. For Pr < 0:1 and low Re (acoustic wave starts to form thin layer of unstable stratification of temperature near upper boundary), as well as for any Pr and Re > 167:7 (the flow is acousticsdominated, basic state weakly depends on boundary conditions type) map of stability and perturbations structure are quite similar qualitatively to that for conductive boundaries. So, presence of acoustic waves once again makes the mode to be of Rayleightype. Situation is complicated by the presence of zone, where acoustic flow is stabilized by convective flow, severing the area of stability in parametric space for high enough Pr (concrete value increases with Re increasing, starting with Pr 0:39). With further Gr growth, flow again becomes unstable. Upper threshold of this area is consequently an analog of ascending branch of stability curve for Re ¼ 0. For lower Pr and Re < 400 there is also area of stability growth with Gr increase, formed by the turn of S-shaped section of lower branch of stability curve. Finally, for 0 < Re < 167:7 and high Pr there is instability zone of acoustic flow, where contributions of convective and acoustic mechanisms into generation of basic flow are approximately equal, perturbation of transversal component of velocity is characterized by the presence of single vortex near upper boundary of layer. When Gr grows further, the flow loses its stability again, the corresponding curve is an analog of stability curve for Re ¼ 0. Critical wavenumbers as functions of Prandtl number are presented in Fig. 14. At low Pr and for lower branch of stability boundary in the case of adiabatic boundary critical k are only slightly smaller, than in the case of conductive boundary. Critical k sharply grows up in the case of the upper branch and adiabatic boundary. In addition to this, in the case of perfectly conductive boundaries, two-dimensional oscillatory Rayleigh-type mode related to the presence of potentially instable temperature stratification areas exists, which is stabilized at quite low Pr or quite high Re. However, it was studied in detail by other researchers earlier, nowhere it is the most dangerous mode and it is absent in the case of adiabatic boundaries, consequently we do not consider it.
4.5. Overall view of stability maps In Figs. 15 and 16 stability maps for both thermal boundary conditions types and different acoustic Reynolds number values are presented. All studied modes are shown including odd threedimensional monotonic mode and two-dimensional Rayleightype mode for perfectly conductive boundaries. In the case of purely convective flow the most dangerous modes sequentially are: transverse rolls, oscillatory longitudinal rolls, ‘even’ monotonic longitudinal rolls. Stability of the latter mode decreases for conductive boundaries and grows for adiabatic boundaries with Prandtl number growth to its high values. In the case of conductive boundaries for Re ¼ 0 at Pr < 0:341 slightly below stability threshold to three-dimensional oscillatory mode, the threshold of stability to inclined waves lies. However, critical inclination angle of wave vector turns to 0 and this mode merges with three-dimensional oscillatory mode when twodimensional shear instability mode is still the most dangerous. The threshold of stability with respect to similar odd mode lies slightly higher than stability threshold with respect to even monotonic three-dimensional mode. For Pr > 0:43 there is also two-dimensional Rayleigh-type mode lying higher than threedimensional modes. In the presence of acoustics two-dimensional and threedimensional oscillatory modes are stabilized, Pr values range, in which three-dimensional mode is more dangerous than the plane one, widens. Mode of inclined waves type for Re–0 exists wherever three-dimensional oscillatory mode exists in a wider Pr range and is more dangerous. However at high enough acoustic wave intensity, as in the case Re ¼ 800 presented in Fig. 17, former even three-dimensional mode is destabilized so significantly that for conductive boundaries three-dimensional monotonic mode becomes the most dangerous immediately after the two-dimensional one. Former three-dimensional odd and two-dimensional Rayleightype modes are stabilized by acoustic wave, more efficiently at high Pr. For adiabatic boundaries the situation is more complicated, in the presence of acoustic wave two-dimensional mode can by dangerous only at low enough Prandtl number values, there is a range where perturbations of inclined waves type are more dangerous, which are replaced by three-dimensional monotonic perturbations.
10 k
10 k
6
6
2 0.001
0.01
0.1
1
Pr 10
2 0.001
(a) Re=0 Re=50 Re=100
0.01
0.1
1
Pr 10
(b) Re=167.7 Re=200 Re=400
Re=600 Re=800 Re=1000
Fig. 14. Critical wavenumber k for three-dimensional monotonic mode as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1056
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
0 Gr
Gr
1x10
5
1x10
5
1x10
3
1x10
3
1x101 0.001
0.01
0.1
1
1x101 Pr 10 0.001
0.01
0.1
(a)
1
Pr 10
(b)
Fig. 15. Maps of stability for perfectly conductive (a) and adiabatic (b) boundaries. Solid line Re ¼ 0, dashed line Re ¼ 100.
0 Gr 1x10
Gr 5
1x10
1x103
5
1x103
1x101 0.001
0.01
0.1
1
1x101 Pr 10 0.001
0.01
0.1
(a)
1
Pr 10
(b)
Fig. 16. Maps of stability for perfectly conductive (a) and adiabatic (b) boundaries. Solid lines Re ¼ 200, dashed lines Re ¼ 800.
1x10 Gr
6
5
1x10 Gr
6
1x10
4
5
3 1x10
4
2
3
1
2
1 4 1x102 0
200
400
600
4 1x102 800 Re 1000 0
(a)
200
400
600
800 Re 1000
(b)
Fig. 17. Maps of stability for perfectly conducting (a) and adiabatic (b) boundaries. Pr ¼ 0:1. 1 two-dimensional mode, 2 mode of inclined waves type, 3 three-dimensional oscillatory mode, 4 three-dimensional monotonous conventionally ‘even’ mode, 5 three-dimensional monotonous conventionally ‘odd’ mode.
In the vicinity of stability curve minimum with respect to the mode of inclined waves type there is yet another similar mode, stability curve of which forms isolated, still more dangerous section. Less dangerous curve for three-dimensional oscillatory mode, lying slightly higher, almost in parallel, splits into two separate sections. Additionally, there is a system of stability zones with respect to three-dimensional monotonous ‘even’ mode. For low Re these are zones of instability of acoustics-dominated flow in the low right corner of the map, for Re > 167:7 these are zones of stability of
convection-dominated flow in the top right corner (see also Fig. 13b). Critical Grashof numbers as function of Reynolds number for Pr ¼ 0:1 are presented in Fig. 17. In the case of conductive boundaries (a) two-dimensional mode, stabilizing with Re growth, is the most dangerous. Mode of inclined wave type is destabilized slightly up to Re 75, later it is also solely stabilized and is the most dangerous in a certain domain. Slightly higher and very close to it there is stability threshold with respect to three-dimensional
1057
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
oscillatory mode, which is always stabilized. However, for high Re the most dangerous is destabilizing three-dimensional monotonic mode even in the case of purely convective flow. Also, higher there is stability threshold with respect to stabilizing three-dimensional monotonic mode, which is odd for Re ¼ 0. In the case of adiabatic boundaries (b) behavior of twodimensional and three-dimensional monotonic ‘odd’ modes is qualitatively similar, yet they are located higher. For low Re mode of inclined waves type is the most dangerous from the very beginning. At Re 530 another, destabilizing branch of this mode appears, and three-dimensional oscillatory mode is replaced by destabilizing branch at Re 450 (see also Figs. 6, 10, 16b). Curve of stability with respect to monotonous ‘even’ mode has S-shaped turn and forms area of stability increasing with Grashof number growth for some values of Re. In the case of perfectly conductive boundaries it is possible to compare our results and [10] with [14]. Despite significant difference in the structure of acoustic flow, created by volumetric force in cross-section of ultrasound beam, two-dimensional mode and three-dimensional oscillatory mode response to the low-intensity acoustic wave in a similar way (they are stabilized), behavior of three-dimensional monotonic mode is also qualitatively similar (it is destabilized). However for low Prandtl number values (Pr < 0:1) and further growth of intensity of acoustic beam acoustic flow under consideration in [14] itself loses stability with respect to two-dimensional perturbations. There is no such effect in our problem, starting from some value of Re, dependence GrðReÞ becomes nearly linear and supposedly this behavior will sustain in the limit Re ! 1. For high Prandtl number values stability maps are qualitatively similar in both problems. Perturbations of inclined wave type were not considered in [14]. 4.6. Energy analyses In order to understand mechanisms, determining stabilization or destabilization of convective flow in the presence of acoustic influence we carried out energy analyses for different modes of instability using the approach described in [14]. Threedimensional oscillatory instability was not considered, because this mode to minimal degree depends on acoustic influence. To obtain equations for energy budget we multiply vector momentum equation for perturbations (its components are Eqs. (14)–(16)) by v and equation for temperature perturbations (17) by h (asterisk means complex conjugation), take real part, and integrate over layer height. Let us define kinetic energy of perturbations as
Z
K¼
vv dy=2;
ð22Þ
y
thermal energy as
Z
E¼
hh dy=2:
ð23Þ
y
Then
@K ¼ Ks þ Kd þ Kb; @t
ð24Þ
@E ¼ Ev þ Eh þ Ed ; @t
ð25Þ
y
Reðv T 00 h Þdy
is a term, describing generation of thermal energy due to vertical R transport of temperature and Eh ¼ y Reðwh Þdy – due to horizonR 2 tal transport, Ed ¼ y Reððh00 k hÞh Þdy=Pr is a contribution of dissipation of fluctuating thermal energy due to conductivity (it is also negative). Terms in momentum equation with pressure give zero contribution after integration. Let us decompose K d in two contributions: K d ¼ K dk þ K d? . Here K dk is a contribution containing components of velocity in the plane in which perturbation develops (plane yOz for twodimensional mode and plane xOy for three-dimensional mode), and K d? is a contribution related to direction perpendicular to that plane. Normalizing kinetic energy equation by jK dk j and denoting normalized terms by strokes, on the stability threshold we have
K 0s þ K 0d? þ K 0b ¼ 1:
ð26Þ
Similarly, normalizing Eq. (25) by jEd j, on the stability threshold we have
E0v þ E0h ¼ 1:
ð27Þ
Finally, in accordance with (11), basic flow can be decomposed to convective and acoustic components, proportional to Gr and Re. Introducing components w0 ¼ wb þ wp ¼ Gr wb0 þ Re wp0 , one can write shear term as K 0s ¼ K 0sb þ K 0sp ¼ Gr K 00sb þ Re K 00sp . Similarly, in accordance with (12), (13) E0v is decomposed into contributions, related to convective and acoustic flow: E0v ¼ E0v b þ E0v p . 4.6.1. Energy analysis for two-dimensional mode Perturbations of temperature are of low importance since this instability mode is related to shear of basic flow, because of that we restrict ourselves by consideration of kinetic energy budget. Perturbations in transversal direction x are absent, consequently K d ¼ K dk , K 0d? ¼ 0. Evolution of different contributions with Reynolds number is shown in Fig. 18. As one can see, buoyancy contribution is always stabilizing. In the case of conductive boundaries it decreases by absolute value up to Re 260, and then remains nearly constant and relatively small. Shear contribution is strongly destabilizing and is balanced mostly by dissipation, until Re 260 it slightly decreases, then its part is practically constant. Acoustic component of basic flow has stabilizing influence, which monotonously grows and starting from Re 260 is more important than buoyancy contribution. Contribution of shear of convective flow has the strongest destabilizing influence, as should be expected, which increases with Re growth. In the case of adiabatic boundaries unstable stratification of temperature is absent, and buoyancy contribution is more important and gradually increases with growth of Re. Destabilizing contribution due to shear of basic flow is more significant, than in the case of conductive boundaries, and increases with growth of Re. Evolution of contribution of shear of acoustic flow is very similar to that in the case of conductive boundaries, but destabilizing influence of shear of convective flow grows faster. In general, acoustic flow acts as stabilizing factor, its influence increases with Re growth. In the case of conductive boundaries at Re > 250 it is the most important stabilizing contribution, in the case of adiabatic boundaries the role of buoyancy contribution due to stable temperature distribution is more important.
R
Reðw00 v w Þdy is a term describing generation of flucR tuating kinetic energy due to shear of basic flow, K d ¼ y Re½u00 u þ Here K s ¼
generation of kinetic energy due to buoyancy, Ev ¼
R
y
v 00 v þ w00 w k ðuu þ vv þ ww Þ dy is a term describing viscous dissipation of fluctuating kinetic energy (it is stabilizing by nature R and consequently it’s negative), term K b ¼ y ReðGrhv Þdy describes 2
4.6.2. Energy analysis for three-dimensional monotonic mode In the case of three-dimensional mode with kz ¼ 0, as follows from (12), K 0b ¼ 1 and destabilizing contribution of buoyancy is balanced by dissipation K dk . In Fig. 19 evolution of different contribution is shown only up to Re ¼ 400, because in the range
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Kinetic energy terms
1058
2
2
1
1
0
0
-1 800 Re 1000 0
-1 0
200
400
600
200
400
(a)
800 Re 1000
600
(b) K'b
K'sb
K's
K'sp
Fig. 18. Evolution of different contributions in energy budget (K 0b , K 0s , K 0sb and K 0sp ) for two-dimensional mode on stability threshold as function of pulsational Reynolds number Re for Pr ¼ 0:1, perfectly conductive (a) and adiabatic (b) boundaries.
100
50
50
0
0
-50
-50
-100
-100
Kinetic energy
100
0
100
200
300
Re 400
0
100
Thermal energy
(a)
K'sb
K's
K'sp
0.8
0.8
0.4
0.4
0
0 200
Re 400
(b)
1.2
100
300
K'd⊥
1.2
0
200
300
(c)
Re 400
0
100
200
300
Re 400
(d) E'v
E'vb
E'h
E'vp
Fig. 19. Evolution of different contributions in energy budget (K 0b , K 0s , K 0sb and K 0sp ) for three-dimensional mode on stability threshold as function of pulsational Reynolds number Re for perfectly conductive (a) and adiabatic (b) boundaries, evolution of contributions in thermal energy budget (E0v , E0h , E0v b , E0v p ) for perfectly conductive (c) and adiabatic (d) boundaries, Pr ¼ 0:1.
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Re ¼ 400 . . . 1000 trends, already visible in range Re ¼ 300 . . . 400, only continue. In the direction of temperature gradient and axis z perturbation due to shear is balanced by dissipation. Until Re 250 in the case of perfectly conductive boundaries these terms are approximately constant, growth of destabilization by acoustic influence is compensated by decreasing of destabilization by shear of convective flow. Later due to sharp decreasing of critical Grashof number basic flow becomes mostly acoustic, its convective component exerts weak stabilizing effect, contribution of acoustic component grows with Reynolds number approximately quadratically. In the case of adiabatic boundaries convective contribution in energy budget is initially stronger and diminishes faster, so that overall contribution of basic flow shear diminishes too, and then becomes practically constant and very weakly stabilizing. Both contributions in thermal energy budget, E0v and E0h , are destabilizing, the last of them is prevailing. In the case of conductive boundaries (Fig. 19c) E0h is initially relatively small and grows with Reynold number, and then after transition to primary acoustic flow is approximately constant. In the case of adiabatic boundaries, it is twofold higher and decreases. Evolution of contribution E0v in both is similar but for adiabatic boundaries it is initially higher. For primary acoustic flow (after Re 240) contribution E0v b is weakly stabilizing. For this mode acoustic influence is strong destabilizing factor, in particular after transition to mostly acoustic flow. In the temperature perturbation most important destabilizing contribution is horizontal transport of temperature, and all destabilizing contributions are balanced almost exclusively by dissipation.
5. Conclusions The effect of thermal boundary conditions on the stability of stationary convective flow in the horizontal layer subjected to the longitudinal temperature gradient and acoustic wave, propagating in the horizontal direction is studied. Stability thresholds of stationary flow with respect to twodimensional, three-dimensional oscillatory, three-dimensional monotonic perturbations and perturbations of inclined waves type are determined. The stabilizing effect of acoustic wave at low Prandtl numbers, when three-dimensional monotonic mode is not the most dangerous, is demonstrated. With the acoustic Reynolds number growth stabilization effect becomes stronger, but the domain of Prandtl number values in which this effect takes place, becomes narrower. At high enough Rayleigh numbers there is a range of Prandtl numbers in which two three-dimensional oscillatory modes coexists in the case of adiabatic boundaries. At high Prandtl number values three-dimensional monotonic perturbations are the most dangerous, stability threshold with respect to them decreases due to acoustic effect. In the case of adiabatic boundaries for high Pr there is a system of stability and instability zones of convection-dominated and
1059
acoustics-dominated flows depending on the contribution of each mechanism of flow generation. It is proved that purely acoustic (Gr ¼ 0) flow is stable with respect to longitudinal rolls in the whole range of Prandtl and Reynolds numbers. Similar result was obtained numerically in [14] for acoustic beam. Its absolute stability with respect to general perturbations looks probable, although is not proved. In [14] purely acoustic flow is found to be unstable to two-dimensional oscillatory perturbations. Energy analysis for two-dimensional and monotonic threedimensional perturbation mode is performed. Influence of different contributions in energy budget is considered. Acknowledgement This research was supported by Russian Science Foundation under Grant No. 14-21-00090. References [1] V.I. Polezhaev, in: H. Freyhardt, G. Muller (Eds.), Growth and Defect Structures, Crystals, Vol. 10, Springer, Berlin Heidelberg, 1984, pp. 87–147. [2] G.Z. Gershuni, P. Laure, V.M. Myznikov, B. Roux, E.M. Zhukhovitsky, On the stability of plane-parallel advective flow in long horizontal layers, Microgravity Q 2 (3) (1992) 141–151. [3] J.E. Hart, Stability of thin non-rotating Hadly circulations, J. Atmos. Sci. 29 (5) (1972) 687–697. [4] G.Z. Gershuni, E.M. Zhukhovitskii, V.M. Myznikov, Stability of a plane-parallel convective flow of a liquid in a horizontal layer, J. Appl. Mech. Tech. Phys. 15 (1) (1974) 78–82. [5] H.P. Kuo, S.A. Korpela, A. Chait, P.S. Marcus, Stability of natural convection in a Shallow cavity, in: 8th Int. Heat Transfer. Conf., San Francisco. Calif, vol. 3, 1986, pp. 1539–1544. [6] A.E. Gill, A theory of thermal oscillations in liquid metals, J. Fluid Mech. 64 (1974) 577–588. [7] G.Z. Gershuni, E.M. Zhukhovitskii, V.M. Myznikov, Stability of plane-parallel convective fluid flow in a horizontal layer relative to spatial perturbations, J. Appl. Mech. Tech. Phys. 15 (5) (1974) 706–708. [8] H.P. Kuo, S.A. Korpela, Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side, Phys. Fluids 31 (1) (1988) 33–42. [9] A.S. Ortiz-Pérez, L.A. Dávalos-Orozco, Convection in a horizontal fluid layer under an inclined temperature gradient, Phys. Fluids 23 (2011) 084107. [10] A.S. Ortiz-Pérez, L.A. Dávalos-Orozco, Convection in a horizontal fluid layer under an inclined temperature gradient for Prandtl numbers Pr > 1, Int. J. Heat Mass Transf. 68 (2014) 444–455. [11] T.P. Lyubimova, D.V. Lyubimov, V.A. Morozov, R.V. Scuridin, H. Ben Hadid, D. Henry, Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number, J. Fluid Mech. 635 (2009) 275–295. [12] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Course of Theoretical Physics, Vol. 6, Butterworth-Heinemann, 1987. [13] D.V. Lyubimov, S.V. Shklyaev, Thermal convection in an acoustic field, Fluid Dyn. 35 (3) (2000) 321–330. [14] W. Dridi, D. Henry, H. Ben Hadid, Stability of buoyant convection in a layer submitted to acoustic streaming, Phys. Rev. E 81 (2010) 056309. [15] D.V. Lyubimov, Thermal convection in an acoustic field, Fluid Dyn. 35 (2) (2000) 177–184. [16] W. Nyborg, Acoustic streaming, Phys. Acoust. 2 (Pt. B) (1965). [17] D.V. Lyubimov, T.P. Lyubimova, V.A. Morozov, Numerical Implementation of a Package for Studying Linear Instability of Non-unidimensional Flows, Bulletin of Perm University. Information Systems and Technologies, 2001. No. 5, pp. 74–81.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
The effect of thermal boundary conditions on a stability of convective flow in a horizontal layer subjected to the longitudinal temperature gradient and acoustic wave T.P. Lyubimova a,b,⇑, R.V. Skuridin a a b
Institute of Continuous Media Mechanics UB RAS, Koroleva Str., 1, 614013 Perm, Russia Perm State University, Bukireva Str., 15, 614990 Perm, Russia
a r t i c l e
i n f o
Article history: Received 19 May 2015 Received in revised form 6 August 2015 Accepted 6 August 2015
a b s t r a c t The effect of thermal boundary conditions on linear stability of stationary convective flow in a horizontal layer subjected to the longitudinal temperature gradient and propagating acoustic wave is studied. Twodimensional shear instability and three-dimensional monotonic and oscillatory instabilities are analyzed. The stability maps for different acoustic Reynolds number values with respect to different perturbation types are obtained. The stabilization under the influence of acoustic wave at low Prandtl numbers is demonstrated. It takes place when two-dimensional shear or three-dimensional oscillatory instability modes are most dangerous. Three-dimensional monotonic mode is destabilized by acoustic influence. Energy analyses for two-dimensional and monotonic three-dimensional modes of instability is carried out. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction The problem of thermal buoyancy convection in the presence of horizontal temperature gradient has multiple practical applications and is widely studied. Such flows appear in the number of technological processes and geophysical phenomena, e.g. some types of flows in the ocean, crust and mantle of the Earth, melt flows in the production of crystals. In the latter case, the character of heat and mass transfer in the melt influences considerably the microstructure of the growing crystal (see [1]). That is why understanding of physical mechanisms, determining dynamics of the melt and its stability is important. In many cases suppression of development of primary instability leads to stabilization of the process in general, which is very desirable. The problem of stability of convective flow in the horizontal layer with rigid perfectly conductive boundaries subjected to the longitudinal temperature gradient is widely studied (see [2,3–7]). As analysis showed, there are several mechanisms of instability. What mechanism is most dangerous, depends on the value of Prandtl number. At low Prandtl numbers Pr < 0:14 twodimensional shear instability mechanism is at work (perturbations are vortices, generated on the boundary of opposing directions flows and look as rolls with axes perpendicular to the gradient of ⇑ Corresponding author at: Institute of Continuous Media Mechanics UB RAS, Koroleva Str., 1, 614013 Perm, Russia. E-mail address: [email protected] (T.P. Lyubimova). http://dx.doi.org/10.1016/j.ijheatmasstransfer.2015.08.019 0017-9310/Ó 2015 Elsevier Ltd. All rights reserved.
temperature). In the Prandtl number range 0:14 < Pr < 0:44 the three-dimensional oscillatory perturbations are most dangerous. These perturbations are internal waves generated in the layer of stable stratification due to energy of basic flow. They represent the rolls with axes parallel to the direction of temperature gradient. At Pr > 0:44 the monotonic three-dimensional modes are the most dangerous. These are perturbations having the form of rolls with axes parallel to the temperature gradient, generated in the area of unstable temperature stratification, i.e. the instability mechanism is of the Rayleigh type. The case of adiabatic boundaries was considered in [8]. It was shown that in this case two-dimensional shear instability is most dangerous in a narrower range of Prandtl number values: at Pr < 0:033. As in the case of perfectly conductive boundaries, with the increase of Prandtl number this instability mode is replaced by three-dimensional oscillatory perturbations which remain the most dangerous at 0:033 < Pr < 0:21. Differ from the case of perfectly conductive boundaries there are no domains of unstable temperature stratification in the basic state temperature profile and due to that the monotonic three-dimensional instability mode of the Rayleigh type does not exist. Instead of that, at Prandtl number larger than 0.21 there arises new instability mode which does not exist for perfectly conductive boundaries. This is monotonic three-dimensional instability mode. The critical Rayleigh numbers for this mode is substantially higher than that for two instability modes discussed above, however due to the sharp stabilization of those modes at Pr 0:033 and Pr 0:21 respectively, at
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Pr > 0:21 the three-dimensional monotonic instability mode is most dangerous. In [9,10] convection in a horizontal fluid layer under inclined temperature gradient was studied. Perfectly conductive boundaries were considered. Linear stability with respect to multiple modes for wide range of Pr was investigated. Analytical expression for the limit Pr ! 1 was obtained. The linear stability of convective flow in the horizontal channel of rectangular cross-section with respect to the perturbation modes similar to perturbations in the horizontal layer, depending on the Prandtl number and channel width, was studied numerically in [11]. It is shown that unidirectional (longitudinal) steady flow in a channel exists only for Pr ¼ 0, at any non-zero Prandtl number values the flow is three-dimensional. Stability maps are obtained, structure of critical perturbations is examined and analyzed. The transition to the limit case of infinitely large width channel (layer) is observed, the qualitative modification of the structure and physical nature of perturbations as channel width diminishes is shown. The presence of large variety of instability mechanisms allows application of different supplementary factors (motion of boundaries, stationary and rotating magnetic fields and others) for control of stationary convective flow stability. One of the promising methods is the use of acoustic wave. It is known (see [12]) that acoustic wave can generate stationary flows in viscous fluid. The effect of plane propagating acoustic wave on the convective flows in the horizontal layer with perfectly conductive boundaries was studied in [13]. The linear stability of stationary flow to twodimensional and three-dimensional perturbations was investigated for low and moderate Prandtl number values. Stability maps for two-dimensional and three-dimensional perturbations are obtained. It is found that the acoustic wave of low enough intensity (small enough values of Reynolds number) makes stabilizing effect; weakly non-linear analysis has shown that all considered types of instability are excited through supercritical bifurcation. The effect of ultrasound beam on the stability of convective flow in the horizontal layer in the case when the layer thickness is much larger than acoustic wave length (which means either larger system dimension or higher sound frequency), was studied in [14]. In this case the beam creates volumetric force, generating Eckart streaming in the fluid. The calculations have shown the presence of stabilizing effect at low and moderate Prandtl number values, significant attention was paid to the influence of the beam width and its position in the layer cross-section. From the point of view of application to the horizontal directional solidification the problem of acoustics effect on the stability of convective flow in horizontal layer with adiabatic boundaries is important. This problem was not studied earlier. Its investigation under assumption of comparable acoustic wavelength and characteristic problem dimension is the goal of the present work. To analyze the effect of thermal properties of layer boundaries we carry out calculations for perfectly conductive boundaries and adiabatic boundaries in parallel.
2. Problem formulation The horizontal layer of fluid, limited from above and from below by rigid horizontal plates located at the distance H from each other is under consideration. Coordinate axis z is directed horizontally and located in-between plates at distance H=2. Axis y is directed vertically upwards, axis x is directed horizontally and perpendicularly to the axis z. Temperature, linearly varying on the longitudinal coordinate, is imposed on the plates: T ¼ Az, where A is a constant. Plane acoustic wave, velocity of which is described by w ¼ axk cosðxðt z=cÞÞ, is propagating in the
1047
direction of temperature gradient and axis z (here a is the amplitude, x is the frequency, c is the velocity of the sound, k is the unit vector of the axis z). The oscillation period is assumed to be small compared to characteristic viscous and thermal time scales. This allows to use averaged description. The governing equations for thermo-acoustic convection were obtained by averaging method in [15]. For the acoustic wave under consideration, due to the spatial uniformity of acoustic field, the equations for average components look similar to the conventional equations of thermal buoyancy convection in the Boussinesq approximation and the effect of acoustic wave which generates the average vorticity in boundary layers near rigid walls of the channel, spreading all over the volume of the fluid due to convection and viscosity and leading to formation of acoustic flow is described by means of effective boundary conditions at the outer border of boundary layer. These boundary conditions are translated directly to the rigid surface, since boundary layer thickness is small. Analysis gives the value v z ¼ w ¼ 3a2 x2 =ð4cÞ for the velocity on the walls of the channel [15,16].
Taking quantities H, m=H, H2 =m, AH and qm2 =H2 as scales for length, velocity, time, temperature and pressure respectively, we write down the system of equations for average momentum, energy and continuity in the form [15]:
@ v =@t þ ðv rÞv ¼ rp þ Dv þ GrT c;
ð1Þ
@T=@t þ ðv rÞT ¼ DT=Pr;
ð2Þ
div
v ¼ 0;
ð3Þ
c is the unit vector of the axis y. The boundary condition for average velocity on the layer boundaries (at y = ±1/2) is
v ¼ Re k
ð4Þ
Additionally, supposing closeness of layer at infinity, we impose the condition of no-flow through cross-section:
Z
1=2
1=2
v z dy ¼ 0:
ð5Þ
In the limit case of perfectly conductive boundaries the boundary condition for temperature is
T ¼ z:
ð6Þ
In the opposite limit case of adiabatic boundaries it is replaced by
@T=@y ¼ 0:
ð7Þ
The problem is characterized by following dimensionless parameters: Gr ¼ gbAH4 =m2 is the Grashof number, Pr ¼ m=v is the Prandtl number, Re ¼ 3a2 x2 H=ð4cmÞ is the pulsational Reynolds number. Here m is the kinematic viscosity, q is the fluid density, g is the gravity acceleration, v is the thermal diffusivity. The problem under consideration permits stationary solution, corresponding to plane-parallel flow, where velocity is uniform along z axis and transversal components are absent. In this case system (1)–(3) reduces to
@p0 =@y þ GrT 0 ¼ 0;
ð8Þ
@p0 =@z þ @ 2 v 0z =@y2 ¼ 0;
ð9Þ
v 0z ¼ @ 2 T 0 =@y2 =Pr:
ð10Þ
Solution for longitudinal component of velocity is a superposition of convective and acoustic flows:
v 0z w0 ¼ Grðy3 y=4Þ=6 þ Reð6y2 1=2Þ:
ð11Þ
1048
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Stationary profile of temperature in the case of perfectly conductive boundaries (problem (8)–(10), (4)–(6)) is
there is a single zone of unstable stratification in the upper part of the layer (Fig. 1f).
T 0 ¼ z þ Re Prðy4 =2 y2 =4 þ 1=32Þ
3. Linear stability problem
þ Gr Prðy5 =120 y3 =144 þ 7y=5760Þ;
ð12Þ
To study stability of basic state we represent the temperature, velocity and pressure fields in the form of sums of basic state and small perturbations, introduce the fields in such a form into original equations and linearize obtained equations with respect to small perturbations. The presence of longitudinal temperature gradient prohibits acquisition of transformation, reducing the problem of three-dimensional perturbations to the problem of two-dimensional perturbations, and in general case it is necessary to consider complete formulation. However, as it is demonstrated below, perturbations, which are of one or another limiting case, are the most dangerous in most areas of parameter space. The coefficients in linearized equations for small perturbations do not depend on time and x and z coordinates (the temperature T 0 depend on z, but its z-derivative, which enters into linearized energy equation for perturbations, is equal to 1). This allows us to consider perturbations of all fields in the form f ðyÞ expðkt þ ikx x þ ikz zÞ. It is convenient to denote kx ¼ k cosðaÞ,
and in the case of adiabatic boundaries (problem (8)–(10), (4), (5), (7)) is:
T 0 ¼ z þ Re Prðy4 =2 y2 =4 þ 1=32Þ þ Gr Prðy5 =120 y3 =144 þ y=384Þ
ð13Þ
Stationary profiles of longitudinal velocity component and temperature with z subtracted at Reynolds number equal to 1000 and different values of Prandtl and Grashof numbers for the cases of perfectly conductive and adiabatic boundaries are presented in Fig. 1. Grashof numbers for this figure correspond to the stability thresholds at given Prandtl number values, and Prandtl numbers are selected in order to illustrate typical structures of the solution. In the case of perfectly conductive boundaries purely convective or convection-dominated flow is characterized by the presence of 0 zones of potentially unstable stratification with @T < 0, @@yq > 0 near @y upper and lower boundaries of the layer, while in its central part 0 temperature distribution does not create instability @T > 0, @y @q @y
kz ¼ k sinðaÞ, kx þ kz k , where a is the angle between wave vector and x axis. Introducing notations u, v , w, h, P for amplitudes of perturbations of x-, y-, z-components of velocity, temperature and pressure respectively we obtain the following spectral-amplitude problem (stroke denotes derivation by y): 2
< 0, Fig. 1a). As relative role of acoustic mechanism of flow gen-
eration increases, Gr number grows, the lower area of potential instability diminishes while the upper grows (Fig. 1b). When acoustic flow dominates, only one area of potential instability remains (Fig. 1c). In the case of adiabatic boundaries convective flow does not generate similar zones of potential instability (Fig. 1d). As contribution of acoustic component of flow increases such zone starts to develop in the upper part of the layer (Fig. 1e). Finally, when the flow becomes acoustics-dominated, situation becomes similar to the case of perfectly conductive boundaries, because acoustic component does not depend on the boundary conditions type,
T0
-0.4 -0.2
0
2
2
ku þ ikz w0 u ¼ ikx P k u þ u00 ;
ð14Þ
kv þ ikz w0 v ¼ P0 k
ð15Þ
2
ð16Þ
kh þ ikz w0 h þ v T 00 þ w ¼ ðh00 k hÞ=Pr;
ð17Þ
2
2
T0
0.2
0.4
-0.4 -0.2
0
T0
0.2
0.4
-40
0.4
0.4
0.2
0.2
0.2
y 0
y 0
y 0
-0.2
-0.2
-0.2
-0.4
-0.4
-0.4
0 v0z
4000
-1000
(a) -0.8 -0.4
0
0 v0z
1000
-1000
0.4
0.8
-0.4 -0.2
0
0.4
-40
0.4
0.4
0.2
0.2
y 0
y 0
y 0
-0.2
-0.2
-0.2
-0.4 0 v0z
(d)
4000
20
0 v0z
40
1000
T0
0.2
0.2
-0.4
0
(c)
T0
0.4
-4000
-20
(b)
T0
v þ v 00 þ Grh;
kw þ ikz w0 w þ w00 v ¼ ikz P k w þ w00 ;
0.4
-4000
2
-20
0
20
40
-0.4 -1000
0 v0z
(e)
1000
-1000
0 v0z
1000
(f)
Fig. 1. Profiles of longitudinal component of velocity (solid lines) and temperature minus z (dashed lines) for Re ¼ 1000, perfectly conductive boundaries ((a) Pr ¼ 0:005, Gr ¼ 288; 830, (b) Pr ¼ 0:0088, Gr ¼ 69; 740, (c) Pr ¼ 1, Gr ¼ 51), and adiabatic boundaries ((d) Pr ¼ 0:002, Gr ¼ 412; 810, (e) Pr ¼ 0:005, Gr ¼ 70; 000, (f) Pr ¼ 1, Gr ¼ 28:08).
1049
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
ikx u þ v 0 þ ikz w ¼ 0; y ¼ 1=2 :
ð18Þ
u ¼ v ¼ w ¼ 0;
Table 1 Critical Grashof number for different modes with different nodes numbers for Pr ¼ 0:09, Re ¼ 1000, adiabatic boundaries.
ð19Þ
Additionally, in the case of perfectly conductive boundaries we have:
y ¼ 1=2 :
h¼0
ð20Þ
Nodes number
51
75
101
201
2D 3D oscillatory 3D monotonic
119,000 15,040 541.2
115,400 15,070 544.4
114,100 15,070 545.7
113,100 15,080 546.8
and in the case of adiabatic boundaries:
y ¼ 1=2 :
h0 ¼ 0;
To estimate grid convergence, Table 1 is provided. As one can see, 101 nodes seem to be adequate.
ð21Þ
Closeness condition (5) for perturbations satisfies automatically if kx ¼ 0. Otherwise flow closure is ensured by periodicity of perturbations in x-direction. The limit case a ¼ p=2 (kx ¼ 0) corresponds to two-dimensional perturbations in the form of rolls with axes perpendicular to basic flow (Fig. 2a), where as seen from (14) u ¼ 0, the limit case a ¼ 0 (kz ¼ 0) corresponds to three-dimensional perturbations in the form of rolls with axes parallel to basic flow (Fig. 2b). In the intermediate case kx –0, kz –0 there are perturbations of inclined waves type. The problem is not changed at simultaneous replacement of y by y, k by k, P by P and Re by Re. At the same time simultaneous replacement of Gr and Re sign doesn’t lead to the physically new situation. Consequently, it is sufficient to consider only the case of positive values of Grashof and Reynolds numbers. Solution of spectral problem at given values of other parameters (Gr, Pr, Re, k, a) allows to obtain the increment k and amplitudes of perturbations as eigenvalues and eigenvectors of the problem. Search of Grashof number value at which real part of k takes zero value gives stability threshold with respect to a given mode. Minimization of this number as function of k (and a in the case of inclined waves) allows to obtain the map of stability of basic flow. Spectral-amplitude problem was discretized by finite difference method of second order of accuracy on the uniform grid with 101 nodes, which reduced it to generalized algebraic problem of eigenvalues AX = k BX, where the matrix A is complex and has a band structure, the matrix B is diagonal, with part of the diagonal elements equal to unity and the others equal to zero, and X is the vector of the unknown perturbations. Computational package developed in [17] was used for the solution which uses the Newton–Raphson method to iteratively find eigenvalues and eigenvectors. Resulting system of linear algebraic equations is solved by LU-decomposition using standard package of linear algebra Lapack. The first guess for eigenvalues and eigenvectors needed for implementation of Newton–Raphson method was obtained by direct numerical simulation of temporal evolution of initial perturbations based on unsteady linearized equations until the evolution is only determined by the most dangerous mode.
4. Numerical results 4.1. Two-dimensional shear instability (kx ¼ 0) Two-dimensional shear instability is related to the development of perturbations in the form of vortices on the boundary of counter streams (transverse rolls). This mode, which is monotonic at Re ¼ 0 (i.e. its imaginary part of k is equal to zero), becomes oscillatory in the presence of acoustic wave. This mode is stabilized by acoustics; stability maps for perfectly conductive and adiabatic boundaries (Fig. 3, logarithmic scale don’t allow to explicitly show data for Pr ¼ 0, but they nearly coincide with the case Pr ¼ 0:001 at the left edge of charts) are similar to each other qualitatively (at Pr ¼ 0 problems coincide). Quantitatively, in the case of adiabatic boundaries as Prandtl number grows, this mode is stabilized more quickly, the flow becomes absolutely stable relative to this mode at Pr 0:32 for perfectly conductive boundaries and at Pr 0:15 for adiabatic boundaries. Approximately twofold difference in maximal Prandtl number values, at which instability with respect to this mode is observed, can be explained by similar (also approximately two-folded) difference in vertical temperature gradient between counter streams, prohibiting development of this mode (see (12) and (13)). Instability of purely acoustic flow (at Gr ¼ 0) to twodimensional shear instability mode was not found, supposedly it is absolutely stable. Graphs of wavenumbers k and frequencies x as function of Prandtl number are plotted in Figs. 4 and 5. They demonstrate qualitatively similar behavior both for kðPrÞ and xðPrÞ for both types of thermal boundaries conditions, just the Prandtl number values at which sharp increase of k and decrease of x occurs in the course of sharp stabilization of this mode with the increase of Pr differs (Pr 0:32 for perfectly conductive boundaries and Pr 0:15 for adiabatic). Critical wavenumbers are bigger for high Pr in the case of adiabatic boundaries, while qualitatively observed pictures are similar. Wavenumbers almost converge at Pr ¼ 0:2 and Pr ¼ 0:1 for considered boundaries conditions variants respectively, and
y
z
x
(a)
(b)
Fig. 2. Schematics of flow structure in the case of two-dimensional (a) and three-dimensional (b) perturbations.
1050
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Gr
Gr
1x10
5
1x104
0.001
0.01
(a)
0.1
1x10
5
1x10
4
Pr
0.001
Re=0 Re=200
0.01
(b)
Re=400 Re=600
Pr
0.1
Re=800 Re=1000
Fig. 3. Maps of stability with respect to two-dimensional shear instability mode of perturbations in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries.
3.2 k
3.2 k
2.8
2.8
2.4
2.4
2 0.001
0.01
(a)
0.1
Pr
Re=0 Re=200
2 0.001
0.01
(b)
Re=400 Re=600
0.1
Pr
Re=800 Re=1000
Fig. 4. Critical wavenumber k for two-dimensional shear instability mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
ω
0
ω
0
-400
-400
-800
-800
-1200 0.001
0.01
(a)
Re=200
0.1
Pr
-1200 0.001
Re=400 Re=600
0.01
(b)
0.1
Pr
Re=800 Re=1000
Fig. 5. Oscillation frequency x for two-dimensional shear instability mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1051
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
then grow. Charts of critical frequencies are also qualitatively similar, frequencies monotonously grow as Re increases. 4.2. Three-dimensional oscillatory mode with kz ¼ 0 Three-dimensional oscillatory perturbations are traveling waves, propagating in the direction perpendicular to the temperature gradient (longitudinal rolls), in the area of stable stratification. Families of stability curves for different Reynolds number values are presented in Fig. 6. At low Reynolds number values stability maps for both variants of thermal boundary conditions are qualitatively similar, at higher Reynolds numbers in the case of adiabatic boundaries their behavior becomes much more complicated. Quantitative differences are high enough even at Re ¼ 0. Let us discuss the results in more detail. For purely convective flow (Re ¼ 0) at low Prandtl numbers critical Grashof number at first decreases as Pr increases (in logarithmic coordinates this dependence is linear and in the limit Pr ! 0 approaches asymptotes lnðGr C Þ 1:05 lnðPrÞ þ 7:15 for perfectly conductive boundaries and lnðGr C Þ 0:494 lnðPrÞþ 7:416 for adiabatic boundaries), reaches minimum (at Pr ¼ 0:288, Gr C ¼ 12; 150 and Pr ¼ 0:103, Gr C ¼ 7332 respectively), then it starts to grow and tends to vertical asymptotes Pr ¼ 0:456 and Pr ¼ 0:207 respectively. At non-zero Reynolds numbers asymptote is not modified as Re grows, but in the vicinity of the stability curve
Gr
Gr θ
w 1x10
minimum the mode is monotonously stabilized with simultaneous decrease of Pr, at which this minimum is attained. For perfectly conductive boundaries this behavior remains at any Re, but in case of adiabatic boundaries at Re > 450 another branch of the same mode appears, which is even destabilized with Re growth in some range of Prandtl numbers ( 0:04 < Pr < 0:15). In the inset of Fig. 6b the structure of both branches of perturbations for Re ¼ 1000, Pr ¼ 0:045, Gr ¼ 18; 550 is presented, on the left for the branch starting in the area of low Prandtl numbers, on the right for the branch which continues in the area of relatively high Prandtl numbers. As it is seen, depending on the boundary conditions type, the structure of temperature perturbations changes drastically. There are two minima on the neutral curves in some zone of parametric space (Fig. 7), and as Prandtl number varies, one or other of them becomes global. All neutral curves intersect in the vicinity of point Gr 19; 000, k 0:85, but apparently there is nothing particular about it. Critical wavenumbers and frequencies are presented in Figs. 8 and 9. While at high Pr there is some qualitative similarity of behavior in two boundary conditions variants considered, at low Prandtl numbers differences are significant, especially at Pr ¼ 0:01 critical frequencies are approximately order of magnitude higher in the case of perfectly conductive boundaries and are decreasing with Pr growth, while in the case of adiabatic boundaries they are almost constant.
w
w
θ
θ
5
1x10
5
w 1x104
θ
w
0.01
1x10 0.1
(a)
Pr
0.01
Re=0 Re=200
θ
4
Pr
0.1
(b)
Re=400 Re=600
Re=800 Re=1000
Fig. 6. Maps of stability relative to three-dimensional oscillatory mode of perturbations in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries. Profiles of longitudinal component of velocity and temperature perturbations are presented in the insets (w, T, solid and dashed lines are real and imaginary parts) at the parametric space points, indicated by arrows.
Gr 21000 20000 19000 18000 17000 0.4
0.8 Pr=0.035 Pr=0.04
Pr=0.045 Pr=0.05
Fig. 7. A family of neutral curves for adiabatic boundaries, Re ¼ 1000.
k 1.2 Pr=0.055
1052
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
2
2
k
k
1.5
1.5
1
1
0.5
0.5
0 0.01
0.1
(a)
Pr
Re=0 Re=200
0 0.01
Pr
0.1
(b)
Re=400 Re=600
Re=800 Re=1000
Fig. 8. Critical wavenumber k for three-dimensional oscillatory mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
In the case of perfectly conductive boundaries at low Pr wavelength of three-dimensional oscillatory perturbations virtually does not depend on Prandtl number, remaining close to k 2, while frequency monotonously grows as Pr diminishes, similar to critical value of Gr. In the case of adiabatic boundaries wavenumber grows up as Pr grows, reaching some maximum. As Pr approaches the threshold of absolute stabilization of the mode for both variants of boundary conditions wavenumber tends to 0 (and wavelength correspondingly to infinity), while frequency tends to some limit value (x ¼ 151 for conductive boundaries and x ¼ 47:6 for adiabatic boundaries). Similar to critical Grashof numbers, this asymptote weakly depends on Reynolds number, i.e. at low and extremely high Prandtl numbers characteristics of perturbations are determined mostly by convective mechanism of flow generation. At Gr ¼ 0 (purely acoustic flow) the exact solution of the problem for perturbations can be obtained, it turns out that the only solution is trivial solution. So, acoustic flow is absolutely stable with respect to longitudinal rolls type instabilities. This conclusion is correct for both monotonic and oscillatory three-dimensional perturbations. Similar result was obtained numerically in [14] for the acoustic beam case.
4.3. Mode of inclined waves type (kx –0, kz –0) Perturbations of inclined waves type can be described as the result of transformation of three-dimensional oscillatory perturbations with kz ¼ 0 when the requirement of orthogonality of wavevector to temperature gradient is removed. Critical angle a in the domain of Pr values where this mode can be the most dangerous does not exceed the value a ¼ 0:3 ¼ 17 (and usually is smaller), so structures of perturbations with kz ¼ 0, kx –0 and with kz –0, kx –0 are similar. Maps of stability (Fig. 10) are also similar (decreasing of critical Gr, as Pr grows, linear in logarithmic coordinates and reaching the minimum, then grows tending to vertical asymptote), however in the case of adiabatic boundaries there are qualitative differences. Stability threshold with respect to this mode is always lower than the threshold with respect to oscillatory longitudinal rolls. At low values of Prandtl and Reynolds numbers (approximately up to Pr 0:01 and Re 250) mode of inclined waves type is destabilized as intensity of acoustic wave grows (very weakly for perfectly conductive boundaries and slightly stronger for adiabatic boundaries). At high values of any of these parameters mode is stabilized. At perfectly conductive boundaries, limiting value of
400
160
ω
ω 120
300 80 200 40
100 0.01
0.1
(a)
Re=0 Re=200
Pr
0 0.01
Re=400 Re=600
Pr
0.1
(b)
Re=800 Re=1000
Fig. 9. Oscillation frequency x for three-dimensional oscillatory mode of perturbations as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1053
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
1x106 Gr
1x106 Gr
1x10
5
1x10
5
1x10
4
1x10
4
0.001
0.01
0.1
Pr
0.001
0.01
(a)
Pr
0.1
(b)
Re=0 Re=200
Re=400 Re=600
Re=800 Re=1000
Fig. 10. Maps of stability with respect to mode of perturbations of inclined waves type in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries.
Prandtl number, at which flow becomes absolutely stable with respect to this mode, is Pr 0:65, about 30% higher than for three-dimensional oscillatory mode with kz ¼ 0. Critical angle of inclination of wavevector here tends to some value a 0:75 ¼ 43 . For adiabatic boundaries instability exists in a wider range of Prandtl number values than for three-dimensional oscillatory mode with kz ¼ 0, stability curve reaches Pr 0:22 (difference from three-dimensional oscillatory mode with kz ¼ 0 is less than 5%), but then declines backwards to lower Pr, approaching the same vertical asymptote Pr ¼ 0:207 as threedimensional oscillatory mode with kz ¼ 0. So, in narrow range of Pr flow is stabilized as Gr grows. Critical wavenumbers and frequencies are plotted in Figs. 11 and 12. In general, the curves are comparable with ones for oscillatory longitudinal rolls. For perfectly conductive boundaries the mode of inclined waves type exists even for Re ¼ 0 if Pr < 0:034 (at higher Prandtl numbers a ¼ 0, mode coincides with three-dimensional oscillatory mode) and is shown in Fig. 8 although merges with curves for the other Re values. For adiabatic boundaries, and Re ¼ 0 these modes coincide for any Pr, splitting occurs only for Re > 0. For adiabatic boundaries at Re > 450 there is additionally specific branch of this mode existing near minimum of stability
2 k
curve, more dangerous than principal branch in some range of Prandtl numbers in the vicinity of Pr ¼ 0:1 and is destabilized as Re grows. Critical wavenumbers and frequencies behavior for the mode under consideration is similar to that for oscillatory longitudinal rolls. Due to qualitative similarity of this mode to three-dimensional oscillatory mode with kz ¼ 0, pure acoustic flow probably is also absolutely stable with respect to it, although it is difficult to strictly demonstrate this statement. 4.4. Three-dimensional monotonic instability mode with kx –0, kz ¼ 0 of Rayleigh origin For Re ¼ 0 there are two three-dimensional monotonic instability modes with kx –0, kz ¼ 0 (longitudinal rolls), which differ from each other in symmetry. Even mode (signs of perturbations of longitudinal component of velocity and temperature are identical near upper and lower boundaries of layer) is more dangerous, and we will restrict ourselves by consideration of it (odd mode lies higher and is stabilized with Re growth). If Re–0, this mode, obviously, does not possess any definite symmetry. For Pr < 0:1 pictures for perfectly conductive and adiabatic boundaries are quite similar,
2 k
1.6
1.5
1.2 1 0.8 0.5
0.4 0 0.001
0.01
0.1
Pr
0 0.001
(a) Re=0 Re=200
0.01
0.1
Pr
(b) Re=400 Re=600
Re=800 Re=1000
Fig. 11. Critical wavenumber k for mode of perturbations of inclined waves type as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1054
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
1000 ω
120 ω
800
100
600
80
400
60
200
40
0 0.001
0.01
20 0.001
Pr
0.1
0.01
(a)
Pr
0.1
(b)
Re=0 Re=200
Re=400 Re=600
Re=800 Re=1000
Fig. 12. Oscillation frequency x for mode of perturbations of inclined waves type as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
for higher Prandtl numbers in the case of adiabatic boundaries stability map significantly complicates, qualitative differences arise. Let us consider at first the case of perfectly conductive boundaries (Fig. 13a). For Re ¼ 0 critical Grashof number monotonously and (in logarithmic coordinates) almost linearly decreases with Prandtl number growth from Gr ¼ 1; 170; 000 for Pr ¼ 0:001 to Gr ¼ 1360 for Pr ¼ 10. Energy of perturbations concentrates mostly near the walls in the area of unstable stratification of temperature (this mode is of Rayleigh-type nature). In the central part of the layer there is a weaker vortex of opposite direction. For non-zero Reynolds numbers and low Pr the mode is weakly destabilized with Re growth. In this case transversal gradient of temperature in the area of unstable stratification in the upper part of the layer grows, while diminishing in the lower part, and correspondingly the energy of perturbations relocates to the upper part of the channel (the upper vortex intensifies and the lower vortex diminishes), while a stagnation zone forms in the lower part. For high Pr it is more correct to speak about destabilization of acoustic flow with Gr growth (flow is mostly generated by acoustic
Gr 1x10
w
θ
5
θ
mechanism on the stability threshold), but despite completely different structure of basic flow, perturbation energy also concentrates in the upper part of the layer in the zone of unstable stratification, although stagnation zone is less pronounced. Stability curves once again are approximately linear in logarithmic coordinates, but lay lower and at a steeper angle and destabilization by acoustic wave is more substantial. In between ranges of ‘low Prandtl numbers’ and ‘high Prandtl numbers’ there is S-shaped transitional section. As Re grows, it translates to lower Pr and becomes steeper. The map of stability to this mode for adiabatic boundaries is presented in Fig. 13b. In the case of adiabatic boundaries area of unstable stratification does not exist due to convective flow. However, as it was shown in [8], at Pr > 0:21 there exists three-dimensional monotonic instability mode with kz ¼ 0, kx –0, which is not related to the existence of the instable temperature stratification. For Re ¼ 0 and low Pr dependence of critical Grashof number on Prandtl number is qualitatively similar to the dependence for perfectly conductive boundaries considered above, with a little bit weaker
Gr
θ
1x10
θ
w
w θ
1x103
w
1x103
θ
θ
w
w
θ
w
w 1
1x10 0.001
w
5
θ
w
θ
1
0.01
0.1
1
1x10 Pr 10 0.001
(a) Re=0 Re=50 Re=100
0.01
0.1
1
Pr 10
(b) Re=167.7 Re=200 Re=400
Re=600 Re=800 Re=1000
Fig. 13. Maps of stability relative to three-dimensional monotonic mode of perturbations in the Pr Gr plane for perfectly conductive (a) and adiabatic (b) boundaries. Profiles of longitudinal component of velocity and temperature perturbations are presented in the insets (w is solid and T is dashed line) in the parameter space points, indicated by arrows.
1055
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
stability, but at Pr 0:15 the minimum Gr c ¼ 130; 000 is reached, and later Gr c increases since the absence of area of unstable temperature stratification leads to mode stabilization. Thus, this mode is not of Rayleigh-type nature and exists in the area where deviation of temperature from conductive distribution is maximal, and convective transfer, which creates it, diminishes. Perturbations concentrate near layer boundaries to even stronger degree, while being almost absent in central part of it. For Pr < 0:1 and low Re (acoustic wave starts to form thin layer of unstable stratification of temperature near upper boundary), as well as for any Pr and Re > 167:7 (the flow is acousticsdominated, basic state weakly depends on boundary conditions type) map of stability and perturbations structure are quite similar qualitatively to that for conductive boundaries. So, presence of acoustic waves once again makes the mode to be of Rayleightype. Situation is complicated by the presence of zone, where acoustic flow is stabilized by convective flow, severing the area of stability in parametric space for high enough Pr (concrete value increases with Re increasing, starting with Pr 0:39). With further Gr growth, flow again becomes unstable. Upper threshold of this area is consequently an analog of ascending branch of stability curve for Re ¼ 0. For lower Pr and Re < 400 there is also area of stability growth with Gr increase, formed by the turn of S-shaped section of lower branch of stability curve. Finally, for 0 < Re < 167:7 and high Pr there is instability zone of acoustic flow, where contributions of convective and acoustic mechanisms into generation of basic flow are approximately equal, perturbation of transversal component of velocity is characterized by the presence of single vortex near upper boundary of layer. When Gr grows further, the flow loses its stability again, the corresponding curve is an analog of stability curve for Re ¼ 0. Critical wavenumbers as functions of Prandtl number are presented in Fig. 14. At low Pr and for lower branch of stability boundary in the case of adiabatic boundary critical k are only slightly smaller, than in the case of conductive boundary. Critical k sharply grows up in the case of the upper branch and adiabatic boundary. In addition to this, in the case of perfectly conductive boundaries, two-dimensional oscillatory Rayleigh-type mode related to the presence of potentially instable temperature stratification areas exists, which is stabilized at quite low Pr or quite high Re. However, it was studied in detail by other researchers earlier, nowhere it is the most dangerous mode and it is absent in the case of adiabatic boundaries, consequently we do not consider it.
4.5. Overall view of stability maps In Figs. 15 and 16 stability maps for both thermal boundary conditions types and different acoustic Reynolds number values are presented. All studied modes are shown including odd threedimensional monotonic mode and two-dimensional Rayleightype mode for perfectly conductive boundaries. In the case of purely convective flow the most dangerous modes sequentially are: transverse rolls, oscillatory longitudinal rolls, ‘even’ monotonic longitudinal rolls. Stability of the latter mode decreases for conductive boundaries and grows for adiabatic boundaries with Prandtl number growth to its high values. In the case of conductive boundaries for Re ¼ 0 at Pr < 0:341 slightly below stability threshold to three-dimensional oscillatory mode, the threshold of stability to inclined waves lies. However, critical inclination angle of wave vector turns to 0 and this mode merges with three-dimensional oscillatory mode when twodimensional shear instability mode is still the most dangerous. The threshold of stability with respect to similar odd mode lies slightly higher than stability threshold with respect to even monotonic three-dimensional mode. For Pr > 0:43 there is also two-dimensional Rayleigh-type mode lying higher than threedimensional modes. In the presence of acoustics two-dimensional and threedimensional oscillatory modes are stabilized, Pr values range, in which three-dimensional mode is more dangerous than the plane one, widens. Mode of inclined waves type for Re–0 exists wherever three-dimensional oscillatory mode exists in a wider Pr range and is more dangerous. However at high enough acoustic wave intensity, as in the case Re ¼ 800 presented in Fig. 17, former even three-dimensional mode is destabilized so significantly that for conductive boundaries three-dimensional monotonic mode becomes the most dangerous immediately after the two-dimensional one. Former three-dimensional odd and two-dimensional Rayleightype modes are stabilized by acoustic wave, more efficiently at high Pr. For adiabatic boundaries the situation is more complicated, in the presence of acoustic wave two-dimensional mode can by dangerous only at low enough Prandtl number values, there is a range where perturbations of inclined waves type are more dangerous, which are replaced by three-dimensional monotonic perturbations.
10 k
10 k
6
6
2 0.001
0.01
0.1
1
Pr 10
2 0.001
(a) Re=0 Re=50 Re=100
0.01
0.1
1
Pr 10
(b) Re=167.7 Re=200 Re=400
Re=600 Re=800 Re=1000
Fig. 14. Critical wavenumber k for three-dimensional monotonic mode as function on Prandtl number for perfectly conductive (a) and adiabatic (b) boundaries.
1056
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
0 Gr
Gr
1x10
5
1x10
5
1x10
3
1x10
3
1x101 0.001
0.01
0.1
1
1x101 Pr 10 0.001
0.01
0.1
(a)
1
Pr 10
(b)
Fig. 15. Maps of stability for perfectly conductive (a) and adiabatic (b) boundaries. Solid line Re ¼ 0, dashed line Re ¼ 100.
0 Gr 1x10
Gr 5
1x10
1x103
5
1x103
1x101 0.001
0.01
0.1
1
1x101 Pr 10 0.001
0.01
0.1
(a)
1
Pr 10
(b)
Fig. 16. Maps of stability for perfectly conductive (a) and adiabatic (b) boundaries. Solid lines Re ¼ 200, dashed lines Re ¼ 800.
1x10 Gr
6
5
1x10 Gr
6
1x10
4
5
3 1x10
4
2
3
1
2
1 4 1x102 0
200
400
600
4 1x102 800 Re 1000 0
(a)
200
400
600
800 Re 1000
(b)
Fig. 17. Maps of stability for perfectly conducting (a) and adiabatic (b) boundaries. Pr ¼ 0:1. 1 two-dimensional mode, 2 mode of inclined waves type, 3 three-dimensional oscillatory mode, 4 three-dimensional monotonous conventionally ‘even’ mode, 5 three-dimensional monotonous conventionally ‘odd’ mode.
In the vicinity of stability curve minimum with respect to the mode of inclined waves type there is yet another similar mode, stability curve of which forms isolated, still more dangerous section. Less dangerous curve for three-dimensional oscillatory mode, lying slightly higher, almost in parallel, splits into two separate sections. Additionally, there is a system of stability zones with respect to three-dimensional monotonous ‘even’ mode. For low Re these are zones of instability of acoustics-dominated flow in the low right corner of the map, for Re > 167:7 these are zones of stability of
convection-dominated flow in the top right corner (see also Fig. 13b). Critical Grashof numbers as function of Reynolds number for Pr ¼ 0:1 are presented in Fig. 17. In the case of conductive boundaries (a) two-dimensional mode, stabilizing with Re growth, is the most dangerous. Mode of inclined wave type is destabilized slightly up to Re 75, later it is also solely stabilized and is the most dangerous in a certain domain. Slightly higher and very close to it there is stability threshold with respect to three-dimensional
1057
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
oscillatory mode, which is always stabilized. However, for high Re the most dangerous is destabilizing three-dimensional monotonic mode even in the case of purely convective flow. Also, higher there is stability threshold with respect to stabilizing three-dimensional monotonic mode, which is odd for Re ¼ 0. In the case of adiabatic boundaries (b) behavior of twodimensional and three-dimensional monotonic ‘odd’ modes is qualitatively similar, yet they are located higher. For low Re mode of inclined waves type is the most dangerous from the very beginning. At Re 530 another, destabilizing branch of this mode appears, and three-dimensional oscillatory mode is replaced by destabilizing branch at Re 450 (see also Figs. 6, 10, 16b). Curve of stability with respect to monotonous ‘even’ mode has S-shaped turn and forms area of stability increasing with Grashof number growth for some values of Re. In the case of perfectly conductive boundaries it is possible to compare our results and [10] with [14]. Despite significant difference in the structure of acoustic flow, created by volumetric force in cross-section of ultrasound beam, two-dimensional mode and three-dimensional oscillatory mode response to the low-intensity acoustic wave in a similar way (they are stabilized), behavior of three-dimensional monotonic mode is also qualitatively similar (it is destabilized). However for low Prandtl number values (Pr < 0:1) and further growth of intensity of acoustic beam acoustic flow under consideration in [14] itself loses stability with respect to two-dimensional perturbations. There is no such effect in our problem, starting from some value of Re, dependence GrðReÞ becomes nearly linear and supposedly this behavior will sustain in the limit Re ! 1. For high Prandtl number values stability maps are qualitatively similar in both problems. Perturbations of inclined wave type were not considered in [14]. 4.6. Energy analyses In order to understand mechanisms, determining stabilization or destabilization of convective flow in the presence of acoustic influence we carried out energy analyses for different modes of instability using the approach described in [14]. Threedimensional oscillatory instability was not considered, because this mode to minimal degree depends on acoustic influence. To obtain equations for energy budget we multiply vector momentum equation for perturbations (its components are Eqs. (14)–(16)) by v and equation for temperature perturbations (17) by h (asterisk means complex conjugation), take real part, and integrate over layer height. Let us define kinetic energy of perturbations as
Z
K¼
vv dy=2;
ð22Þ
y
thermal energy as
Z
E¼
hh dy=2:
ð23Þ
y
Then
@K ¼ Ks þ Kd þ Kb; @t
ð24Þ
@E ¼ Ev þ Eh þ Ed ; @t
ð25Þ
y
Reðv T 00 h Þdy
is a term, describing generation of thermal energy due to vertical R transport of temperature and Eh ¼ y Reðwh Þdy – due to horizonR 2 tal transport, Ed ¼ y Reððh00 k hÞh Þdy=Pr is a contribution of dissipation of fluctuating thermal energy due to conductivity (it is also negative). Terms in momentum equation with pressure give zero contribution after integration. Let us decompose K d in two contributions: K d ¼ K dk þ K d? . Here K dk is a contribution containing components of velocity in the plane in which perturbation develops (plane yOz for twodimensional mode and plane xOy for three-dimensional mode), and K d? is a contribution related to direction perpendicular to that plane. Normalizing kinetic energy equation by jK dk j and denoting normalized terms by strokes, on the stability threshold we have
K 0s þ K 0d? þ K 0b ¼ 1:
ð26Þ
Similarly, normalizing Eq. (25) by jEd j, on the stability threshold we have
E0v þ E0h ¼ 1:
ð27Þ
Finally, in accordance with (11), basic flow can be decomposed to convective and acoustic components, proportional to Gr and Re. Introducing components w0 ¼ wb þ wp ¼ Gr wb0 þ Re wp0 , one can write shear term as K 0s ¼ K 0sb þ K 0sp ¼ Gr K 00sb þ Re K 00sp . Similarly, in accordance with (12), (13) E0v is decomposed into contributions, related to convective and acoustic flow: E0v ¼ E0v b þ E0v p . 4.6.1. Energy analysis for two-dimensional mode Perturbations of temperature are of low importance since this instability mode is related to shear of basic flow, because of that we restrict ourselves by consideration of kinetic energy budget. Perturbations in transversal direction x are absent, consequently K d ¼ K dk , K 0d? ¼ 0. Evolution of different contributions with Reynolds number is shown in Fig. 18. As one can see, buoyancy contribution is always stabilizing. In the case of conductive boundaries it decreases by absolute value up to Re 260, and then remains nearly constant and relatively small. Shear contribution is strongly destabilizing and is balanced mostly by dissipation, until Re 260 it slightly decreases, then its part is practically constant. Acoustic component of basic flow has stabilizing influence, which monotonously grows and starting from Re 260 is more important than buoyancy contribution. Contribution of shear of convective flow has the strongest destabilizing influence, as should be expected, which increases with Re growth. In the case of adiabatic boundaries unstable stratification of temperature is absent, and buoyancy contribution is more important and gradually increases with growth of Re. Destabilizing contribution due to shear of basic flow is more significant, than in the case of conductive boundaries, and increases with growth of Re. Evolution of contribution of shear of acoustic flow is very similar to that in the case of conductive boundaries, but destabilizing influence of shear of convective flow grows faster. In general, acoustic flow acts as stabilizing factor, its influence increases with Re growth. In the case of conductive boundaries at Re > 250 it is the most important stabilizing contribution, in the case of adiabatic boundaries the role of buoyancy contribution due to stable temperature distribution is more important.
R
Reðw00 v w Þdy is a term describing generation of flucR tuating kinetic energy due to shear of basic flow, K d ¼ y Re½u00 u þ Here K s ¼
generation of kinetic energy due to buoyancy, Ev ¼
R
y
v 00 v þ w00 w k ðuu þ vv þ ww Þ dy is a term describing viscous dissipation of fluctuating kinetic energy (it is stabilizing by nature R and consequently it’s negative), term K b ¼ y ReðGrhv Þdy describes 2
4.6.2. Energy analysis for three-dimensional monotonic mode In the case of three-dimensional mode with kz ¼ 0, as follows from (12), K 0b ¼ 1 and destabilizing contribution of buoyancy is balanced by dissipation K dk . In Fig. 19 evolution of different contribution is shown only up to Re ¼ 400, because in the range
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Kinetic energy terms
1058
2
2
1
1
0
0
-1 800 Re 1000 0
-1 0
200
400
600
200
400
(a)
800 Re 1000
600
(b) K'b
K'sb
K's
K'sp
Fig. 18. Evolution of different contributions in energy budget (K 0b , K 0s , K 0sb and K 0sp ) for two-dimensional mode on stability threshold as function of pulsational Reynolds number Re for Pr ¼ 0:1, perfectly conductive (a) and adiabatic (b) boundaries.
100
50
50
0
0
-50
-50
-100
-100
Kinetic energy
100
0
100
200
300
Re 400
0
100
Thermal energy
(a)
K'sb
K's
K'sp
0.8
0.8
0.4
0.4
0
0 200
Re 400
(b)
1.2
100
300
K'd⊥
1.2
0
200
300
(c)
Re 400
0
100
200
300
Re 400
(d) E'v
E'vb
E'h
E'vp
Fig. 19. Evolution of different contributions in energy budget (K 0b , K 0s , K 0sb and K 0sp ) for three-dimensional mode on stability threshold as function of pulsational Reynolds number Re for perfectly conductive (a) and adiabatic (b) boundaries, evolution of contributions in thermal energy budget (E0v , E0h , E0v b , E0v p ) for perfectly conductive (c) and adiabatic (d) boundaries, Pr ¼ 0:1.
T.P. Lyubimova, R.V. Skuridin / International Journal of Heat and Mass Transfer 91 (2015) 1046–1059
Re ¼ 400 . . . 1000 trends, already visible in range Re ¼ 300 . . . 400, only continue. In the direction of temperature gradient and axis z perturbation due to shear is balanced by dissipation. Until Re 250 in the case of perfectly conductive boundaries these terms are approximately constant, growth of destabilization by acoustic influence is compensated by decreasing of destabilization by shear of convective flow. Later due to sharp decreasing of critical Grashof number basic flow becomes mostly acoustic, its convective component exerts weak stabilizing effect, contribution of acoustic component grows with Reynolds number approximately quadratically. In the case of adiabatic boundaries convective contribution in energy budget is initially stronger and diminishes faster, so that overall contribution of basic flow shear diminishes too, and then becomes practically constant and very weakly stabilizing. Both contributions in thermal energy budget, E0v and E0h , are destabilizing, the last of them is prevailing. In the case of conductive boundaries (Fig. 19c) E0h is initially relatively small and grows with Reynold number, and then after transition to primary acoustic flow is approximately constant. In the case of adiabatic boundaries, it is twofold higher and decreases. Evolution of contribution E0v in both is similar but for adiabatic boundaries it is initially higher. For primary acoustic flow (after Re 240) contribution E0v b is weakly stabilizing. For this mode acoustic influence is strong destabilizing factor, in particular after transition to mostly acoustic flow. In the temperature perturbation most important destabilizing contribution is horizontal transport of temperature, and all destabilizing contributions are balanced almost exclusively by dissipation.
5. Conclusions The effect of thermal boundary conditions on the stability of stationary convective flow in the horizontal layer subjected to the longitudinal temperature gradient and acoustic wave, propagating in the horizontal direction is studied. Stability thresholds of stationary flow with respect to twodimensional, three-dimensional oscillatory, three-dimensional monotonic perturbations and perturbations of inclined waves type are determined. The stabilizing effect of acoustic wave at low Prandtl numbers, when three-dimensional monotonic mode is not the most dangerous, is demonstrated. With the acoustic Reynolds number growth stabilization effect becomes stronger, but the domain of Prandtl number values in which this effect takes place, becomes narrower. At high enough Rayleigh numbers there is a range of Prandtl numbers in which two three-dimensional oscillatory modes coexists in the case of adiabatic boundaries. At high Prandtl number values three-dimensional monotonic perturbations are the most dangerous, stability threshold with respect to them decreases due to acoustic effect. In the case of adiabatic boundaries for high Pr there is a system of stability and instability zones of convection-dominated and
1059
acoustics-dominated flows depending on the contribution of each mechanism of flow generation. It is proved that purely acoustic (Gr ¼ 0) flow is stable with respect to longitudinal rolls in the whole range of Prandtl and Reynolds numbers. Similar result was obtained numerically in [14] for acoustic beam. Its absolute stability with respect to general perturbations looks probable, although is not proved. In [14] purely acoustic flow is found to be unstable to two-dimensional oscillatory perturbations. Energy analysis for two-dimensional and monotonic threedimensional perturbation mode is performed. Influence of different contributions in energy budget is considered. Acknowledgement This research was supported by Russian Science Foundation under Grant No. 14-21-00090. References [1] V.I. Polezhaev, in: H. Freyhardt, G. Muller (Eds.), Growth and Defect Structures, Crystals, Vol. 10, Springer, Berlin Heidelberg, 1984, pp. 87–147. [2] G.Z. Gershuni, P. Laure, V.M. Myznikov, B. Roux, E.M. Zhukhovitsky, On the stability of plane-parallel advective flow in long horizontal layers, Microgravity Q 2 (3) (1992) 141–151. [3] J.E. Hart, Stability of thin non-rotating Hadly circulations, J. Atmos. Sci. 29 (5) (1972) 687–697. [4] G.Z. Gershuni, E.M. Zhukhovitskii, V.M. Myznikov, Stability of a plane-parallel convective flow of a liquid in a horizontal layer, J. Appl. Mech. Tech. Phys. 15 (1) (1974) 78–82. [5] H.P. Kuo, S.A. Korpela, A. Chait, P.S. Marcus, Stability of natural convection in a Shallow cavity, in: 8th Int. Heat Transfer. Conf., San Francisco. Calif, vol. 3, 1986, pp. 1539–1544. [6] A.E. Gill, A theory of thermal oscillations in liquid metals, J. Fluid Mech. 64 (1974) 577–588. [7] G.Z. Gershuni, E.M. Zhukhovitskii, V.M. Myznikov, Stability of plane-parallel convective fluid flow in a horizontal layer relative to spatial perturbations, J. Appl. Mech. Tech. Phys. 15 (5) (1974) 706–708. [8] H.P. Kuo, S.A. Korpela, Stability and finite amplitude natural convection in a shallow cavity with insulated top and bottom and heated from a side, Phys. Fluids 31 (1) (1988) 33–42. [9] A.S. Ortiz-Pérez, L.A. Dávalos-Orozco, Convection in a horizontal fluid layer under an inclined temperature gradient, Phys. Fluids 23 (2011) 084107. [10] A.S. Ortiz-Pérez, L.A. Dávalos-Orozco, Convection in a horizontal fluid layer under an inclined temperature gradient for Prandtl numbers Pr > 1, Int. J. Heat Mass Transf. 68 (2014) 444–455. [11] T.P. Lyubimova, D.V. Lyubimov, V.A. Morozov, R.V. Scuridin, H. Ben Hadid, D. Henry, Stability of convection in a horizontal channel subjected to a longitudinal temperature gradient. Part 1. Effect of aspect ratio and Prandtl number, J. Fluid Mech. 635 (2009) 275–295. [12] L.D. Landau, E.M. Lifshitz, Fluid Mechanics, second ed., Course of Theoretical Physics, Vol. 6, Butterworth-Heinemann, 1987. [13] D.V. Lyubimov, S.V. Shklyaev, Thermal convection in an acoustic field, Fluid Dyn. 35 (3) (2000) 321–330. [14] W. Dridi, D. Henry, H. Ben Hadid, Stability of buoyant convection in a layer submitted to acoustic streaming, Phys. Rev. E 81 (2010) 056309. [15] D.V. Lyubimov, Thermal convection in an acoustic field, Fluid Dyn. 35 (2) (2000) 177–184. [16] W. Nyborg, Acoustic streaming, Phys. Acoust. 2 (Pt. B) (1965). [17] D.V. Lyubimov, T.P. Lyubimova, V.A. Morozov, Numerical Implementation of a Package for Studying Linear Instability of Non-unidimensional Flows, Bulletin of Perm University. Information Systems and Technologies, 2001. No. 5, pp. 74–81.