Stability of density stratified flow in the boundary layer1

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Journal of Hydrodynamics Ser.B, 2006,18(5): 505-511

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STABILITY OF DENSITY STRATIFIED FLOW IN THE BOUNDARY LAYER* XIE Ming-liang, LIN Jian-zhong State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China China Jiliang University, Hangzhou 310018, China, E-mail:[email protected]

(Received June 16, 2005) ABSTRACT: The stability of the laminar flat plate boundary layer is investigated numerically by solving the linear Orr–Sommerfeld equations for the disturbulence amplitude function. These equations include the terms of viscosity, density stratification, and diffusion. Neutral stability curve and the critical Re numbers are computed for various Richardson (Ri) numbers and Schmidt (Sc) numbers. The results show that the larger the Ri, the larger the critical Re for Sc < 10 . The flow is stable for Ri < 0 , when Sc is very small or the mass diffusion coefficient is very large. But for Ri > 0 , the effects of diffusion are reversed for Sc < 10 . For Sc > 10 , the critical Re rapidly decreases to zero as the Sc increases for a given Ri number. The critical Re rapidly decreases as the Ri increases. KEY WORDS: stability, stratified flow, boundary layer, blasius flow

1. INTRODUCTION The influence of vertical density variations on the stability of flow past a flat horizontal wall, is in a sense related to the case of centrifugal forces acting on a homogeneous fluid flowing along a curved wall. When the arrangement is stable, the density decreases upward, , whereas it becomes unstable when the variation in density is reversed. In the case of flow with stable density stratification, turbulent mixing in the vertical direction is impeded because heavier particles must be lifted and lighter particles must be depressed against hydrostatic forces. Turbulence can even be completely suppressed if the density gradient is strong enough, the phenomenon being of some importance in certain meteorological and hydrodynamic processes [1-6].

The theory of stability of laminar flows decomposes the motion into a mean flow and a disturbance superimposed on it. The investigation of the stability of such a disturbed flow can be carried out using either of the two methods, i.e., energy method and small disturbances method. The initial stage of the transition process in the Blasius boundary layer on flat plate is a problem that has received much attention, both theoretically and practically. In this stage of the transition process, the study is focused on the stability of small, two-dimensional periodic disturbances in the boundary layer. Prandtl analyzed the phenomena connected with density gradients using the energy method. He has shown that the stability of stratified flows depends on the stratification parameter, Ri, in addition to the usual dependence on Re. The energy method used by Richardson has shown that turbulence may disappear at Ri > 2 . Taylor refined Prandtl’s reasoning and obtained Ri ≥ 1 as the limit of stability. Taylor and Goldstein were the first to apply the method of small disturbances to this problem. The effects of stratification on the stability of parallel shear flows are usually studied in an inviscid approximation (see, for example, Howard[7]). The problem is then governed by the Taylor–Goldstein equation, which similar to Rayleigh’s stability equation for a homogeneous fluid, is singular at any point where the basic flow speed U ( y ) is equal to the reciprocal wave speed

* Project supported by the National Natural Science Foundation of China (Grant No: 10372090). Biography: XIE Ming-liang (1974-), Male, Ph.D. Student Corresponding author: LIN Jian-zhong, E-mail: [email protected]

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C . A statically stable density distribution would be expected, on general physical grounds, to have a stabilizing effect, and it has been proved by Miles and Howard that stability is assured if the local Ri exceeds 1/4 everywhere. Schlichting investigated the stability of flows with density stratification by applying Tollmien’s theory. The calculation was based on the assumption of a Blasius profile for a flat plate with a density gradient in the boundary layer and constant density outside it. It was found that the critical Re rapidly increased as the Ri increased, resulting in a change in its value from 645 for Ri = 0 (homogeneous flow) to infinity for Ri = 1/24. Thus the flow remains stable everywhere on the flat plate for Ri > 1/24. Lin et al. [8] derived the spatial stability equation of moving jet containing dense suspended solid particles by means of the continuum phase-coupled model. The stability curves of moving jet for different downstream distances, Reynolds number of flow-field, particle properties, and velocities of jetting device were obtained. It was found that the positive velocity of jetting device widens the unstable frequency range of flow field but the effect of the negative velocity is contrary to that of positive velocity. In addition, particles existing in the flow field restrain the instability of flow field, and the restraining effect increased with the decrease in the Reynolds number of flow field. You and Lin[9] carried out a linear stability analysis on the circular pipe flow of fiber suspensions. The constitutive equation for the fiber suspensions was set up and the modified Orr–Sommerfeld equation was derived. The results show that the fiber additives will enhance the flow stability. Wan et al.[10] studied a linear instability of the Taylor–Couette flow between two rotating coaxial cylinders in the presence of the semi-concentrated fibers. On the basis of the model of an anisotropic fluid described by Erickson, a set of modified stability equations were derived by introducing small disturbance to basic flow. The numerical solution of hydrodynamic instability shows that the fiber additives have a stabilizing effect on the flow and this effect is more obvious for higher values of fiber volume fraction, fiber aspect ratio, and lower values of two-cylinder radius ratio. The equations for the perturbation lead to the Orr–Sommerfeld equations with small disturbulence methods. An attempt to numerically calculate the characteristic functions φ ( y ) of the Orr–Sommerfeld equations for a large set of prescribed pairs of values of the wavelength α , and Re, puts enormous demands on the capacity of a computer. This explains why Tollmien and Schlichting investigated this problem yet again and adopted a very tedious analytic procedure. Details of these calculations can be found in the books of Lin[11], and Yih[12] gave a summary of the hydrodynamic stability of the flow of

nonhomogeneous fluids (stratification). The first successful numerical solution of the Orr–Sommerfeld equation was given by Kurtz and Crandall[13] and was improved by Jordinson[14]. White[15] gave a summary of the difficulties associated with the numerical solution of the Orr–Sommerfeld equation. Although the stability of stratified shear flows has been studied for many years, the early investigations were usually concerned with specific flow configurations. This article deals with the stability of the laminar Blasius flow with continuously varying velocity and density profiles. Numerical methods have been used to find the spectrum of eigenvalues of the Orr–Sommerfeld equations. It is emphasized that the eigenvalues found are those of the finite difference approximation.

2. GOVERNING EQUATIONS For a two-dimensional incompressible mean flow and an equally two-dimensional disturbance, the Navier–Stokes equations are:

⎛ ∂u ∂u ∂u ⎞ ∂p +u +v ⎟ = − + ∂x ∂y ⎠ ∂x ⎝ ∂t

ρ⎜

⎛ ∂ 2u ∂ 2u ⎞ μ⎜ 2 + 2 ⎟ ⎝ ∂x ∂y ⎠

(1)

⎛ ∂v ∂v ∂v ⎞ ∂p + u + v ⎟ = −ρ g − + ∂x ∂y ⎠ ∂y ⎝ ∂t

ρ⎜

⎛ ∂ 2v ∂ 2v ⎞ + 2⎟ 2 ⎝ ∂x ∂y ⎠

μ⎜

(2)

where u , v are the velocity components, p is the pressure, and ρ is the density of solution. The viscosity μ is assumed to be constant. The diffusion equation is:

⎛ ∂u ∂v ⎞ ∂s ∂s ∂s +u + v + s⎜ + ⎟ = ∂t ∂x ∂y ⎝ ∂x ∂y ⎠

⎛ ∂2s ∂2s ⎞ + 2⎟ 2 ⎝ ∂x ∂y ⎠

κ⎜

(3)

where s is the concentration of the solution, and k is the mass diffusivity. The variation of density with concentration is given by

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ρ = ρ 0 [1 + a ( s − s0 )]

⎛ ∂ 2 u ′ ∂ 2u ′ ⎞ + 2⎟ 2 ∂y ⎠ ⎝ ∂x

(4)

where a is the coefficient, and a = 1/ ρ 0 . If diffusion of the solute is taken into account for a dilute solution, the equation of continuity takes the form

μ⎜

⎛ ∂v′ ∂v′ ∂V ∂v′ ∂V ⎞ +U + u′ +V + v′ ⎟+ ∂x ∂x ∂y ∂y ⎠ ⎝ ∂t

ρ⎜

⎛ ∂u ∂v ⎞ ∂ρ ∂ρ ∂ρ +u +v +ρ⎜ + ⎟= ∂t ∂x ∂y ⎝ ∂x ∂y ⎠

⎛ ∂2s ∂2s ⎞ + 2⎟ 2 ⎝ ∂x ∂y ⎠

κ⎜

⎛ ∂V ∂V ∂V ⎞ +U +V ⎟ = − ρ ′g − t x y ∂ ∂ ∂ ⎝ ⎠

ρ′⎜

u = U + u ′ , v = V + v′ , p = P + p ′ , (7)

In most cases, it is assumed that the quantities related to the disturbance are small compared with the corresponding quantities of the main flow. Substituting Eq. (7) into Eqs.(1)–(2) and neglecting the quadratic terms in the disturbance velocity component, we obtain

⎛ ∂U ∂U ∂U ⎞ ∂p′ +U +V + ⎟=− ∂x ∂y ⎠ ∂x ⎝ ∂t

ρ′⎜

V

⎛ ∂u ′ ∂v′ ⎞ ∂ ( ρ ′ − s′) + (ρ − S) ⎜ + ⎟+ ∂y ⎝ ∂x ∂y ⎠

v′

⎛ ∂U ∂V ⎞ ∂( ρ − S ) + ( ρ ′ − s′) ⎜ + ⎟=0 ∂y ⎝ ∂x ∂y ⎠

(6)

Let the mean flow, which may be regarded as steady, be described by its Cartesian velocity components, U, V, and its pressure P, density ρ ,and concentration S. The corresponding quantities for the nonsteady disturbance will be denoted by u′ , v′ , p ′ , ρ ′ , and s′ , respectively. Hence, in the resultant motion, we have

⎛ ∂u′ ∂u′ ∂U ∂u′ ∂U ⎞ ρ⎜ +U + u′ +V + v′ ⎟+ ∂x ∂x ∂y ∂y ⎠ ⎝ ∂t

(9)

∂ ( ρ ′ − s′) ∂ ( ρ ′ − s′) ∂( ρ − S ) +U + u′ + ∂t ∂x ∂x

∂( ρ − s) ∂( ρ − s) ∂( ρ − s) +u +v + ∂t ∂x ∂y

ρ = ρ + ρ ′, s = S + s′

⎛ ∂ 2 v′ ∂ 2 v′ ⎞ ∂p′ +μ⎜ 2 + 2 ⎟ ∂y ∂y ⎠ ⎝ ∂x

(5)

From Eqs. (3)–(5), it follows that

⎛ ∂u ∂v ⎞ ( ρ − s) ⎜ + ⎟ = 0 ⎝ ∂x ∂y ⎠

(8)

∂s′ ∂s′ ∂S ∂s′ ∂S +U + u′ + V + v′ + ∂t ∂x ∂x ∂y ∂y ⎛ ∂u ′ ∂v′ ⎞ ⎛ ∂U ∂V S⎜ + + ⎟ + s′ ⎜ ⎝ ∂x ∂y ⎠ ⎝ ∂x ∂y

⎛ ∂ 2 s′ ∂ 2 s′ ⎞ κ⎜ 2 + 2 ⎟ ∂y ⎠ ⎝ ∂x

(10)

⎞ ⎟= ⎠ (11)

The flow in the boundary layer can be regarded as a good approximation to the parallel flow Assuming the mean flow with U = U ( y ) , V = 0 , and a density profile of the form S S0 = e −by = 1 − by when by is small, where b is a constant. As far as the pressure in the main flow is concerned, it is obviously necessary to assume dependence on x as well as on y , because the pressure gradient ∂P ∂x maintains the flow. We assume that P = P ( x ) + ρ g ( H − y ) and the pressure gradient ∂P ∂y = − ρ g . The two-dimensional perturbation is expressed in terms of the stream function and the density function:

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ψ ( x, y, t ) = φ ( y )ei(α x − β t ) = φ ( y )eiα ( x −Ct )

ρ ′ = s′ = − −

(12)

dρ ϕ ( y )ei(α x − β t ) = dy

dρ ϕ ( y )eiα ( x −Ct ) dy

∂ψ = φ ′( y )ei(α x − β t ) = φ ′( y )eiα ( x −Ct ) ∂y

v′ = −

(U − C )( D 2 − α 2 ) + D 2U (13)

(14)

∂ψ = −iαφ ( y )ei(α x − β t ) = ∂x

−iαφ ( y )eiα ( x −Ct )

(15)

where the parameters α , β , and C represent the wave number, frequency, and phase velocity of perturbation, respectively. It is possible to make the Boussinesq approximation if the total density change across the shear layer is small when compared with the average density,. This involves neglecting all density variations in the equations of motion except in the gravitational term. The amplitude of the perturbation is assumed to be small and the substitution of ψ and ϕ in the linearized vorticity equation for the perturbation leads to the ordinary, fourth-order Orr–Sommerfeld equations after the elimination of pressure:

β (U − )(φ ′′ − α 2φ ) − φU ′′ = α

ν (4) g dρ (φ − 2α 2φ ′′ + α 4φ ) − ϕ iα ρ dy

(16)

β κ )ϕ = (ϕ ′′ − α 2ϕ ) iα α

(17)

φ + (U −

The equations of motion are made dimensionless using the free steam velocity, U 0 , the displacement thickness of the Blasius boundary layer,

δ = vx U 0 , and the kinematics viscosity, v . Then L4φ + Riϕ = 0

(19)

L4 ≡ (iα Re) −1 ( D 4 − 2α 2 D 2 + α 4 ) −

from Eq.(12) it is possible to obtain the components of the perturbation velocity:

u′ =

φ − L2ϕ = 0

(18)

(20)

L2 ≡ (iα ReSc ) −1 ( D 2 − α 2 ) − (U − C )

(21)

where D=d/dy, Schmidt numbers, Sc = ν / k , Reynolds numbers, Re = U 0δ /ν , Froude numbers

1 dρ ρ dy 2 Richardson number Ri = J Fr .

Fr = U 0 2 gδ

,

J =−

and

over-all

The boundary conditions follow from the fact that the perturbation velocities disappear at the wall and far out in the mainstream. The first condition immediately leads to

φ = Dφ = 0 and ϕ = 0 at y = 0

(22)

For the outer boundary conditions, the equation for the position far from the wall is given as:

( D 2 − α 2 ) 2 − iRe(α − β )( D 2 − α 2 ) = 0

(23)

The required solution that fits the outer boundary condition is evidently:

φ = Ae−α y + Be−γ y where

A,

B

(24)

are

arbitrary constants, and + iRe(α − β ) . For the values of α , C

γ =α and Re relevant to the problem, it is evident that γ >> α so that e −α y << e − γ y for y > 0 . Hence 2

2

the relevant condition can be expressed in the form φ ~ e −α y for large values of y. In the analytical solutions, this condition is applied at the edge of the boundary layer, but at y = 8.8 in the present calculation. That is：

φ = Dφ = 0 and ϕ = 0 at y = 8.8

(25)

In the time-amplified case considered here, it is assumed that α , Sc , Ri , and Re are real and known, and the problem is that of finding a complex eigenvalue C with a corresponding eigenvector φ .

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3.

NUMERICAL PROCEDURE First, the differential equation is replaced by a set of different equations, which is referred to as the algebraic model. A transformation is applied to the function φ , the truncation errors in the expressions

preceding section are shown in Fig. 1. The critical Re number of homogeneous fluids for Blasius boundary layer is 530, using the program. It is a slightly larger than the result (520) of Jordinson[14], but the number n =44 is approximately half of the number in his program. Thus the program is feasible for the calculation.

φ and D 2φ being thus reduced to Ο (h 6 ) 4 and Ο ( h ) , respectively, here h is the step length

for D

4

used in the numerical integration.

( D 2φ ) j = ( D 4φ ) j =

φ j +1 − 2φ j + φ j −1 h2

φ j + 2 − 4φ j +1 + 6φ j − 4φ j −1 + φ j − 2 h4

(26)

(27)

The algebraic model can be expressed in the matrix form. Second, an iterative technique is used to find the eigenvalues of the matrix. A computer program was written in MATLAB to perform the iteration. It is necessary to make a suitable choice of the step length h used in the calculations. h is, in practice, fixed by choosing the number n of equal intervals into which the range 0 ≤ y ≤ 8.8 is subdivided, so that h = 8.8 n . Jordinson used n = 80 and the program was adapted to function with n = 44 for the main calculation; this value represents the best compromise between the competing demands of accuracy and of the solution of the Blasius equation solved by Howarth.

RESULTS AND DISCUSSIONS The result of such a calculation for a prescribed laminar flow can be graphically represented in an α -Re diagram because every point of this plane corresponds to a pair of values of Real (C) and Image (C). In particular, the locus Image (C) = 0 separates the region of stable from that of unstable disturbances. This locus is called the curve of neutral stability. The point on this curve at which the Re number has its smallest value is of the greatest interest since it indicates the value of the Re number below which all individual oscillations decay, whereas above which at least some are amplified. This smallest Re number is the critical Reynolds number or limit of stability with respect to the type of laminar flow under consideration. 4.1. Validation of the numerical methods The results of stability calculations performed in accordance with the method described in the

Fig.1 Curves of neutral stability for a two-dimensional boundary layer with two-dimensional disturbances for homogeneous fluid

4.2

Diffusion for homogeneous fluid When Ri=0 the equations of stability have a simple form L4φ = 0 and φ − L2ϕ = 0 or L4 L2 j = 0 . If the diffusion is neglected, the equations reduces to L4φ = 0 which is the Orr-Sommerfeld equation. The critical Re numbers are 550 and 530, respectively. The diffusion has a stable effect on the hydrodynamic stability. The result is given in Fig. 1.

4.

Fig.2 Neutral stability curves for various Ri number

4.3 Effects of Richardson number for the flow of stratification Figure 2 is the neutral stability curves for various Ri numbers, and the relationship between Ri number and the critical Re number is shown in Fig.3. It was

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found that the critical Re number rapidly increases as the Ri number increases for a given Sc number (Sc < 10 ). The larger Sc number, the more rapidly the critical Re number increases. It is interesting, and quite surprising, that even at Ri < 0 , there is a stable mode. But as the absolute value of the Ri number increases, the critical Re number rapidly decreases to zero.

and Ri > 0 . For Ri < 0 , the reverse occurs. And for Ri = 0 , the critical Re number is a constant (550). But for Sc > 10 , the critical Re number rapidly decreases to zero as the Sc number increases for a given Ri number, and the larger the Ri number, the more rapidly the critical Re number decreases.

Fig.5 Neutral stability curves for various Sc number Fig.3 Relationship between Re (critical) and Ri for various Sc

For Sc > 10 , the relationship between the critical Re and Ri becomes more intricate, which can be found in Fig.4.

5.

CONCLUSIONS The effects of stratification on the hydrodynamic stability of the Blasius plate boundary layer flow are presented in this article. A parametric analysis shows that for Sc < 10 , the larger Ri number, the larger the critical Re number is, which agrees with the theoretical analysis given by Schlichting. The stability mode for Ri < 0 , when Sc number is very small or the mass diffusion coefficient is very large is also obtained. But for Ri > 0 , the effects of diffusion are reversed under Sc < 10 . For Sc > 10 , the critical Re number rapidly decreases to zero as the Sc number increases for a given Ri number. The larger Ri number, the more rapidly the critical Re number decreases.

REFERENCES Fig.4

Relationship between critical Re and Sc for various Ri

4.4

Effects of Schmidt number The influence of Sc number on the critical Re number is shown in Fig. 4. Figure 5 shows the neutral stability curves for various Sc numbers. It can be found that the critical Re number is 550 when Sc → 0 , which is the reason why the Orr–Sommerfeld equations becomes L4φ + Riϕ = 0 and φ − L2ϕ = 0 or ScL2 L4φ = 0 when Sc → 0 . It was also found that the critical Re number rapidly increases as the Sc number increases for a given Ri number under the conditions of Sc < 0.1

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