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Structure of Reynolds shear stress in the central region of plane Couette flow
FLUID DYNAMICS RESEARCH ELSEVIER
Fluid Dynamics Research 18 (1996) 65-79
Structure of Reynolds shear stress in the central region of plane Couette flow K n u t H. B e c h , H e l g e I. A n d e r s s o n *
Department of Applied Mechanics, Thermodynamics and Fluid Dynamics, Norwegian University' of Science and Technoloqy, N-7034 Trondheim, Norway Received 27 March 1995; revised 22 December 1995; accepted 29 January 1996
Abstract The structures and mechanisms for maintaining a high Reynolds shear stress throughout the core region of plane turbulent Couette flow were examined by means of databases originating from a direct numerical simulation. At the relatively low Reynolds number considered, the mean shear rate and some one-point statistics exhibited a non-negligible variation in the core region, which conflicted with the postulated homogeneity. The slope of the mean velocity profile was below a theoretically established lower bound for the limit of infinite Re. Analysis of the time-averaged and structural information obtained by conditional sampling showed that mean shear generation and velocity-pressure gradient correlations played a crucial role in the generation and annihilation of the Reynolds shear stress. It was moreover observed that strong, very localized velocity fluctuations in the wall-normal direction were essential in both processes. A simple conceptual model was proposed to explain the physical significance of the pressure field associated with quadrant 2 events.
I. Introduction In spite o f being a classical problem in fluid mechanics, the shear-driven motion o f an incompressible fluid bounded by two walls in relative motion is o f practical interest in a variety o f contexts, ranging from lubrication technology to thin-film coating. The simplest flow o f this kind is the plane Couette flow, i.e. the flow between parallel planes in which no average pressure gradient exists in the flow direction. The fluid flow is maintained solely by the uniform motion o f the planes, and the Reynolds number (based on the relative wall speed) determines whether the flow is in the laminar or turbulent regime. The fully developed turbulent plane Couette flow is a paradigm o f turbulent shear flows, in which the total shear stress dU
=
*Corresponding author.
0169-5983/96/$15.00 @ 1996 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved PIlSO 1 6 9 - 5 9 8 3 ( 9 6 ) 0 0 0 0 8 1
(1)
66
K.H. Bech, H.I. Andersson/Fluid Dynamics Research I8 (1996) 65 79
is constant (and equal to its wall value zw) throughout the entire flow ~. Moreover, since the mean velocity vector V = ( U ( y ) , 0 , 0 ) is aligned with the walls, the motion is homogeneous in planes parallel to the bounding walls, i.e. the mean value of any combination of the turbulent fluctuations becomes a function of the single coordinate (y) measuring the distance from the channel centre. The fully developed Couette flow is therefore particularly amenable to theoretical analysis (von Karman, 1937; Busse, 1970; Lund and Bush, 1980; Missimer and Thomas, 1983; Gersten, 1985; Schneider et al., 1990; Afzal, 1993) and phenomenological modelling (see, e.g. Andersson and Pettersson (1994) for references to one-point closure modelling). In spite of its fundamental nature, however, the plane Couette flow is difficult to realize in the laboratory, and the number of reliable experimental studies are accordingly scarce. The pioneering laboratory investigation is that of Reichardt (1956, 1959) whereas detailed turbulence statistics were provided by E1 Telbany and Reynolds (1982) and more recently by Aydin and Leutheusser (1991) and Tillmark and Alfredsson (1993). Further statistics and structural information have been made available by direct numerical simulations (DNSs) (Lee and Kim, 1991; Papavassiliou, 1993; Kristoffersen et al., 1993) and an elaborate comparison between near-wall data from both numerical and physical experiments was made by Bech et al. (1995). The aforementioned studies have revealed some prominent features of turbulent Couette flow, namely that the near-wall regions are practically the same as in pressure-driven Poiseuille flow and flat-plate boundary layers, whereas the central region closely resembles that of homogeneous turbulent shear flow. The latter part of this statement is deliberately vague since some conflicting hypotheses still prevail in the literature. While the core region is undoubtedly anisotropic due to its non-vanishing Reynolds shear stress, it still remains an open question whether the core flow is homogeneous in the wall-normal direction or not. Under the assumption of constant shear stress -puT, yon Karman (1937) postulated the existence of "homologous" turbulence, i.e. "the correlation coefficients are independent of the location of the point considered" (although they are not invariant as far as rotation and reflection are concerned). Moreover, based on the assumed constancy of -puT, von Karman derived a formula for the mean shear rate, viz., dU/dy cx 1/cos(~y +/~), where and fl are integration constants. According to Eq. (1), however, a uniform mean shear rate is inevitably implied by the assumed constancy of the Reynolds shear stress. The variation of the mean velocity deduced by von Karman therefore conflicts with the underlying assumption of "homologous" turbulence. Later theoretical studies have also led to conflicting conclusions. By means of a variational method, Busse (1970) found an extremalizing solution for plane Couette flow, which exhibited a constant mean shear rate. A comparison with the experimental data of Reichardt (1959) showed that the measurements reflected the theoretically established lower bound Uw/4h on the mean shear rate dU/dy in the interior of the plane Couette flow. On the contrary, Lund and Bush (1980) showed through the use of matched asymptotic expansions that the mean velocity gradient becomes vanishingly small in the core region as the Reynolds number tends to infinity. This contrasts with the measurements by Aydin and Leutheusser (1991), which showed that the logarithmic part of the velocity profiles extended right to the centre of their Couette channel.
l ln the following, U is the mean (streamwise) velocity and u, v, w are the fluctuating velocity components in the streamwise (x), wall-normal (y) and spanwise (z) directions, p and p denote the constant fluid viscosity and density, respectively.
K.H. Bech, H.I. Andersson/ Fluid Dynamics Research I8 (1996) 65 -79
67
The present study focuses on the nearly homogeneous core region of the turbulent Couette flow. The non-vanishing turbulence production makes this part of the flow fundamentally different from that of plane Poiseuille flow, in which not only the turbulence production but also the Reynolds shear stress vanish at the centre plane. The objective of the investigation is therefore twofold, namely: (a) to explore the dynamics of the cross-correlation ~-~ (i.e. the turbulent shear stress), and (b) to examine the postulated homogeneity of the central region. This will be accomplished by means of statistical (long-time averages) and structural (conditional sampling) information deduced from the direct simulation reported by Bech et al. (1995).
2. The numerically generated databases and sampling techniques The present DNS was carried out utilizing the computer code ECCLES by Gavrilakis et al. (1986). The incompressible momentum equations are discretized by second-order accurate central differences. The Poisson equation for the pressure is Fourier-transformed with respect to the streamwise and spanwise homogeneous directions, and the resulting equations are solved directly for each time step. The time advancement is implemented through the Adams-Bashforth scheme. No-slip boundary conditions are used for the velocity components at the walls, while periodicity is assumed in the xand z-direction. Consider plane Couette flow where the parallel walls move in opposite directions. The lower wall located at y = - h propagates with velocity -Uw and the upper wall at y --- h moves with Uw in the streamwise (x) direction. The bulk Reynolds number, Re = Uwh/v, was 1300, where v = l~/P is the kinematic viscosity. The Reynolds number based on friction velocity (u, = x/~w/P) became Re~ = u~h/v = 82.2. The size of the computational box was 10nh x 2h x 4nh in the streamwise, wall-normal and spanwise directions, respectively. The corresponding number of grid points was 256 x 70 x 256. The computational mesh was non-uniform in the y-direction, the grid point next to the walls being at y+ ~- 0.35, whereas the largest spacing y+ -~ 3.9 was at the centreline. For comparison, the equally distributed grid points in the homogeneous xz plane yielded Ax + -~ 10.1 and Az + --~ 4.0. The time step was kept constant and equal to O.03v/u2~. This simulation supercedes the DNS reported by Kristoffersen et al. (1993) with respect to grid resolution and box size. A comparison between the two simulations was made by Bech and Andersson (1994), while a more complete description of the new simulation was given by Bech et al. (1995), in which detailed comparisons between computed and measured turbulence statistics are provided. The statistical results presented here will be taken from two different databases. The time-averaged statistics were sampled during 16.4h/u~ and constitute the first database (the statistics database). The second, which was used for conditional sampling and calculation of auxiliary terms, consisted of 12 fields of velocity and pressure, i.e. a complete description of the flow in the computational box at 12 different time levels. The time separation between the fields was O.15h/u~. The latter data will be referred to as the field database. The total sampling time for the field database was about 1.65h/u~, which should be adequate. Because of the relative short time interval between the fields, one event could have been detected several times during its evolution. This is also the case for the time-averaged statistics, and was not considered as a problem as long as a sufficient number of different events was detected.
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K.H. Bech, H.L Andersson/Fluid Dynamics Research 18 (1996) 65-79
In Section 3.2, results on conditional sampling will be presented. Three different detection schemes were tested in the plane y --- 0, i.e. the centre plane. The first scheme, which will not be reported here, detected regions with large wall-normal momentum. The second was the quadrant splitting detection due to Lu and Willmarth (1973), where it was demanded that u and v should be in a certain quadrant in the uv-plane and that uv/V~ should exceed some threshold level tQ2. Only events in the second quadrant (Q2) were considered here. The last scheme applied detected events with high instantaneous production of -uv, i.e. V2(Cgu/Oy) p
v2(dU/dy)
~>tpQ2 and u < 0,v > 0,
(2)
where tpQ2 is some threshold level. Note that only quadrant 2 events were considered, hence the abbreviation PQ2 (production by Q2-events) will be applied for the scheme described by Eq. (2).
3. Results In this section, results on the statistical properties as well as the dynamics of the turbulent shear stress will be presented. The first subsection deals with time-averaged statistics, the second is concerned with conditional sampling and ensemble averages. The time-averaged quantities provide definite information concerning -~-0. The ensemble averages yield information on the kinematics of events involved in the generation and destruction of the negative correlation between u and v. In this section, and throughout the rest of the paper, all variables have been non-dimensionalized by the channel half-width h and the friction velocity u,. In order to keep the notation as simple as possible, the non-dimensional variables are not distinguished from the corresponding dimensional variables in the first part of this paper. It might be argued that u, is not a typical velocity in the central region. An alternative choice in a region which is close to local equilibrium is q / = (eh) 1/3, where e is the dissipation rate of turbulent kinetic energy. Assuming local equilibrium, u,/#i is approximately proportional to RG-1/3 and thus near order unity for small Reynolds numbers. In order to rescale the present results, the factor u~/~ = 0.63 (for y = 0) should be applied.
3.1. Time-averaged dynamics of the turbulent shear stress The results presented in this subsection will be taken from the statistics database. The overbar denotes averaging in homogeneous m-planes and time. Note that data from the two half-channels were averaged and that the wall is at y = - 1 and the centreline at y = 0. The first figure visualizes the terms in Eq. (1). The curves were found to be smooth and the sum of the turbulent and viscous shear stresses was constant within 0.2% from wall to channel centre. This constancy indicates statistically converged results. The stress curves intersect at y = -0.872 or y+ = 10.5 where the maximum production of turbulent kinetic energy (Pk = -V~dU/dy) occurs. The partition among the viscous and turbulent contributions to the total shear stress varied substantially across the flow. At the centreline, for example, the viscous stress Re~ l dU/dy and the turbulent stress - ~ contributed 4.5% and 95.5%, respectively. Fig. la shows that - ~ remained practically constant (to within 1%) throughout the central region, i.e. y > -0.4, thereby supporting
K.H. Bech, H.I. Andersson/FluidDynamicsResearch18 (1996) 65-79
69
a)
1.2
,
,
~
,
,
, - ~ - - ~ ,
,
b) 0.l
,
....
, ....
1
o
0.075 0.8
o o
\
o o \\
0.6
0.05
0.4 0.025 0.2 0
/,
,
,
i
i
-0.8
-1
-0.6
i
-0.4
I
0
-0.2
0
-
~ t
-0.5
,
,
,
-0.25
,
0
Y Fig. 1. (a) Viscous and turbulent shear stress from wall to centreline. (b) Viscous shear stress in the central region compared with various theories. ( - - ) -~-f; ( - - - ) (- - -) -riO+ (r-l) -fi-f,,~; (o) logarithmic profile, 1/(~c(1 + y)Re,), ~c = 0.284; ( x ) von Karman theory, cos[roy/2]), k = 0.446. k and x was found from at the centreline.
Re~-]dU/dy; (~z/2)/(kRe,
dU/dy
Re~-]dU/dy;
a) 8
~
4 I ...
,
,
i
b) i
. . . .
,
"
i
"""
-... :,..~',-~::~..::~,
0
~~~'..~-.-.--4
- i
-t
l" /
-8
i~'
-8
-4
0 U
i
i
4
i
i
i
8
-8
-4
0
4
8
U
Fig. 2. Quadrant plots in the uv-plane at (a) y = - 0 . 5 and (b) y=0. Broken lines denote fourth grid point at one instant are shown•
uv I=
10. Values at every
the frequently assumed constancy of-fi-~. The mean shear parameter S = d(U/U~,)/d(y/h) was 0.22 at the centreline. A close-up of the viscous shear stress Re[ldU/dy away from the walls is shown in Fig. lb. The corresponding variation deduced from a logarithmic velocity profile, i.e. dU/dy = 1/to(1 + y), as well as the variation derived from the analytical formula proposed by von Karman (1937), i.e. dU/dy = (½n)/kcos(½nY), have also been included. The constants ~c and k were obtained from the mean shear at the centreline where dU/dy = SRe/Re~ = 3.52. It is readily observed that the solution deduced from the logarithmic profile, as advocated by Aydin and Leutheusser (1991), seriously exaggerates the variation of the mean shear, whereas von Karman's proposal compares favourably with the DNS data in the core region. Quadrant plots of (u, v) are presented in Fig. 2. Extremely strong negative correlations between u and v were found to exist. The fluctuations were of approximately the same strength at the centreline
70
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79
and half way towards the wall. In the turbulent boundary layer, fluctuations with [ uv I exceeding 4.5 were characterized as large by Lu and Willmarth (1973), while the broken lines in Fig. 2 represent I uv ]= 10. The probability density distribution of the fluctuating Reynolds shear stress was very different from the Gaussian with a skewness factor of about 4.5 and flatness (kurtosis) slightly exceeding 20. The latter is consistent with the flatness factor at y+ ~- 80 in the Poiseuille flow simulation of Kim et al. (1987), while their skewness factor was about 3. This suggests that uv events are equally intermittent in Poiseuille and Couette flow, whereas the probability density function is more asymmetric in the latter flow in which structures with significant turbulent shear stress contribute to the mean 2. The dynamics of the turbulent shear stress -VO is described by the following non-dimensional transport equation, e.g. Hinze (1975):
8t8(_fi_~)~dU=v ~y + -dyUVd'+~"2~ypud(~?u_ P~y + P~xOV)-v~--~2y2fi-O+2v~ avcgxk = Pl2 + d/12+ dP2 + ~b,2 + d~2 - ~,2,
(3)
where the symbolic notation for the various terms has been introduced for convenience. Because -V~ was of primary concern in the present study, the signs in Eq. (3) have been inverted as compared to Hinze's dynamic equation for the correlation ~-~. In the statistically steady state considered here, ~ ( - V ~ ) / ~ t = 0 and the delicate balance between the different terms on the right-hand side of Eq. (3) is presented in Fig. 3a. The observed variations of the various terms in the near-wall region are qualitatively the same as in the plane Poiseuille flow investigated by Mansour et al. (1988). As in Poiseuille flow, the complexity of the -~T-budget is reduced in the central region, both with respect to the number of significant terms and their variation. The dominating contributions were the production P~2 and the pressure-strain correlations ~bl2, which both exhibited a modest monotonically reduction towards the centreline. The viscous diffusion d~2 was practically negligible for y > -0.5, while the viscous dissipation rate e12 attained an approximately constant level El2 < 0.01 in the central 40% of the channel. Likewise, the turbulent diffusion due to velocity (dtl:) and pressure (dP2) fluctuations remained constant in the core region (y > -0.4). The constancy of d~e and dr2 revealed that the covariances uv 2 and pu varied linearly with y in the core region of the Couette flOW.
It is noteworthy that d~2 exhibited the same behaviour as the diffusion of turbulent kinetic energy, to which - d ( u Z v ) / d y was the dominating contribution. The strong positive correlation between -uZv and uv: is obviously due to the negative correlation between u and v away from the walls. This moreover suggests that there was no fundamental difference in the turbulent transport in the ydirection of the shear stress -k--~ compared to the normal stress u 2. The dissipation was found to be a relatively modest sink term except in the vicinity of the walls (~12/P12 = 0.125 at y = 0) compared to its dominating role in the budget for the turbulent kinetic energy where e/Pk = 1.175 at y = 0. This could be expected since e~2 consists of relatively weak cross correlations between velocity gradients as opposed to e = v(OujSxk)(SuJOxk). 2Here, it should be recalled that the skewness of uv inevitably goes through zero at the symmetryline (y+ = 180) in the Poiseuille flow simulation.
K, H. Bech, H. L Andersson / Fluid Dynamics Research 18 (1996) 65-79
71
a)
b _ l t l l j r
0.1
0.05
0
0
-0.1
-0.05 •
-1
-0.8
~
,
-0.6
,
I
-0.4
J
,
,
[
,
-0.2
y
,
I
J
0
-0.5
t
r
I
i
]
I
-0.25
I
i
r
i
I
0
y
Fig. 3. Terms in the transport equation for - ~ f . (a) Conventional splitting, all terms shown from wall to centreline. (b): Close-up of central region, pressure terms added. (- - -) dt12; (--) d~'2; ( - - - - - - ) de2; ( - - - ) -~12; ( x ) ~bl2; (o) P12; ( . ) dP2 + q~12; ( . . . . ) sum of terms (scarcely visible). Non-dimensionalized with u~/v.
The most important sink term in Eq. (3) was the pressure-strain term ~D12. Unlike the diagonal elements of ~bij, which redistribute energy among the normal Reynolds stress components, the offdiagonal terms (i # j ) are not redistributive. This suggests that the present decomposition of the pressure-velocity interactions, as advocated recently by Groth (1991), does not necessarily ease a sound physical interpretation. Here, it was found convenient to contract terms involving pressure fluctuations to a velocity-pressure gradient term
I-I,2 -= ~12 + dP2 -~ U~y + V~x.
(4)
This further simplifies the dynamics of - ~ , as shown in the close-up of the central region in Fig. 3b. A more detailed examination of the different pressure terms can be made from Fig. 4, in which the individual contributions to ~/)12 and H12 have been plotted. None of the terms, which all acted as sinks in Eq. (3), turned out to be negligible. The major contribution to H~2 in the near-wall region was uOp/~y, but this term gradually decreased towards the centre where v~p/t3x dominated. It is moreover noteworthy that the variation of vOp/t3x coincides with that of -pOv/Ox. This simply reflects the fact that the streamwise pressure diffusion d(~-~)/dx should vanish in fully developed flows. The equivalence of the independently computed terms vt3p/Ox and -p~3v/~x in Fig. 4 therefore supports the adequacy of the present sampling time.
3.2. Structures obtainedfrom conditional samplin9 Ensemble averages obtained from conditional sampling may give an impression of the length scales and structure-characterizing events that are important for the dynamics of the turbulent shear stress.
72
K.H. Bech, H.L Andersson/Fluid Dynamics Research 18 (1996) 65-79
0.02
...
~
. ~
• ~
0
. o
x
o
o
•
o
o
~
°
o
-0.02 -0.04 -0.06 -0.08 -1
-0.8
-0.6
-0.4
-0.2
0
Y Fig. 3. Pressure terms in the transport equation for - E l . (o) - p 8u/Oy; (x) - p Ov/~3x; ( - - ) u ~?p/~3y; (- - -) v c3p/dx. Non-dimensionalized as in Fig. 3. Bech et al. (1995) showed that near-wall internal shear layers detected by the VISA 3 technique contributed substantially to -V~. These events were, however, confined to the near-wall region and not associated with the high level of the Reynolds shear stress in the central region. The mean shear is substantially higher in the near-wall region than in the central region, implying that the turbulent structure near the centreline is different from the more universal near-wall structure. In fact, VISA-detection was also conducted in the central region, but the detected events were weak and very scarce. So detections around the centreline should preferably be made with a method that triggers directly on - u v rather than on some specific structure. The detection algorithms, as described in Section 2, were applied to the field database. For the purpose of Q2 detections, a threshold value to2 --- 10 was adopted. From Fig. 2 it can be inferred that relatively few events with extreme negative correlations between u and v were selected. The events detected at one instant in the xz-plane at y = 0 were plotted in Fig. 5a. The number of islands (detections) in the plane was 41 4. In Fig. 5b, the events detected with the PQ2 scheme at the same instant can be seen, the number of PQ2 islands being 294. The threshold level was tpQ2 = 20, and therefore only regions with extremely strong instantaneous production were taken into account. The rationale for examining extreme events was that a spatial structure would not be blurred by the ensemble averaging. Some of the events detected simultaneously by both the Q2 scheme and the PQ2 scheme were marked with arrows. There was no one-to-one correspondence between strong negative correlation between u and v and strong instantaneous production of the correlation. The areas detected with the Q2 algorithm were generally larger than the PQ2 islands, probably because the latter scheme detects partly on gradients which tend to have shorter length scales than the variables themselves. The ensembles were generated by sampling flow variables in volumes centred around the maximum of each event. That is, u, v, w and p in regions of size 7.24 x 2 x 2.41 centred around the peak of each island in Fig. 5 were averaged. This procedure was followed for each of the 12 fields in the database, and one ensemble was generated by each of the detection schemes. 3Variable interval space averaging. 4The number is an average over the field database.
K.H. Bech, H.I. Andersson / Fluid Dynamics Research 18 (1996) 65 -79
73
a) 10o
0
,
b)
0
10
-~,,,
o °',k,
" "
o~
0
.
0
.
.
.
,
.
°
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.
10
.
,
20
~.
.
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.
r
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X
Fig. 4. Events detected by (a) the Q2-scheme and (b) the PQ2-scheme. The contour levels correspond to the detection threshold levels. Some of the events detected by both schemes are pointed out by arrows. Averages of the ensemble (denoted by angular brackets) obtained with Q2 detection can be studied in Fig. 6. The diagrams, showing a cross-section of the ensemble averages at z = 0, give some information on the length scales of the extreme events considered. The corresponding ensemble averages in the xz-plane (not shown here) gave length scales that compared well (within 5%) with length scales obtained from the spanwise two-point correlations for u and v derived from the statistics database. A further observation from the two-point correlations was that v had the same length scales in the homogeneous directions x and z. Because of the observed agreement between time-averages and ensemble averages with respect to length scales, it was assumed that the conditional sampling procedure captured the essential features of the events and not only a fraction of them. It is interesting to note that (u) in Fig. 6a extends out to the walls, i.e. events at y = 0 have correlation lengths e( h. The outer contours are sligthly deformed by the mean shear. Only topographically simple velocity structures can be seen, i.e. {u) and {v) exhibit a single minimum and maximum, respectively, thus yielding a maximum in - { u v - V~) in Fig. 6c. It is evident that the Q2 scheme detects events with a wide span of length scales and that the ensemble average will not provide elaborated structures. Besides, averages based on one-point detection schemes cannot reproduce spatial asymmetries of the individual events. The pressure contours in Fig. 6d did, however, exhibit an interesting asymmetry, assumingly because only Q2 (and not Q4) events were considered. The Q2 events may be thought of as obstructions in the mean flow, i.e. lumps of fluid travelling with lower speed than the surrounding fluid and therefore giving rise to high-pressure upstream and low-pressure downstream of the lump. This pictorial model is partly confirmed by the pressure contours and also by plots of ensemble averages in the xzplane. Corresponding ensemble averages obtained with the PQ2 scheme are displayed in Fig. 7. The structures detected by the two different schemes differed mostly with respect to the streamwise fluctuating velocity. The PQ2 ensemble was more asymmetric with respect to y = 0 because strong instantaneous gradients ~?u/~?y were a dominant feature of the detected events. The higher number of
74
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79 1.0
0.0
-I.0 1.0
t .
:::::::::::::::::::::::::::::::::
......
a)
............ ~.";;::~-:.!:~.--%;...i-:-~:U". . . . . . . . ' ,.-" . ....... ~,,:':!:i:!;!?(~.!~:~,:,", "," ", . . . . . - - . - . . , . ,.:.,~,,,.'.f~,,.,.;.,,,:,: . . . :
,o I
-1.0 1.0
."
,'""..._
b)
. c)
o.o] -1.0
:::,' 0.0 -1.0
"::-~ .... '~
d)
.,'
........
-4.0-3.0-2.0-1.0
0.0 1.0 2.0 3.0 4.0
Fig. 5. Ensemble averages obtained with Q2-detection. (a) (u), contour increment 0.25; (b) (v), contour increment 0.25; (c) - ( u v - ~-6), contour increment 1.0; (d) (p), contour increment 0.5. Broken lines correspond to negative values. Non-dimensionalized with u~. 1.0
o.oI
a)
.......... , ....:-?,~:~¢~/~.:.-:.................. .. ......... ::::: I~::<" ....... :;;::"
I.O 0.01
b) ~
o
-i.0 1.0
c)
-1.0 1.0
d)
o.oI
1o"0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
Fig. 6. Ensemble averages obtained with PQ2-detection. Caption as in Fig. 6.
pressure contours obtained with the PQ2-detection was probably due to the fact that the events exhibited strong u-velocity gradients, and these gradients contribute to the pressure distribution through the integrated Poisson equation for the pressure. By applying ensemble averaging instead of time averaging, a modified version of Eq. (3) can be obtained. The number of terms originating from the non-linear terms of the Navier-Stokes equations
K.H. Bech, H.I. Andersson/FluidDynamics Research 18 (1996) 65-79 1.0 0.0 -1.0 1.0 0.0
] ]
75
a)
b) Q?.
-1.0 " ".... :: ,;2 {:" :i.--..:"-
'°i
c)
0.0
-1.0 '~ " ,', ,;G" ':---,--,-;--::..: :/-. -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 X
Fig. 7. Conditionally averaged production terms obtained from Q2-detection. (a) (v2)dU/dy; (b) ~(Ou/~?y};(c) ~@v/~?x}. Contour increment 0.05. Non-dimensionalized as in Fig. 3. will increase. The resulting production terms for - ( u v ) are listed below: dU
/~U)
/0V)
/~?U)
/0V)
The first three terms on the right-hand side, as detected by the Q2-scheme, are visualized in Fig. 8. The last two terms are not shown because they were found to be less influential and without any distinct features. Note that these two terms can be combined into -Fg(Ow/~z) due to mass continuity. The spanwise velocity fluctuations w, as well as their gradients, were found to be relatively small after inspection of ensemble averages in the xy- and xz-planes. The production terms in Eq. (5) appear as products of one time-averaged and one ensemble-averaged quantity. The latter averages to zero in the long-time limit in the last four terms, while the former can be treated as a constant for a given y. The first term in Eq. (5) gave the strongest contribution to (Plz) because of rather intense v fluctuations. As can be inferred from Fig. 6b, (?v/?x I was substantial around the centre of the events, thereby resulting in high magnitude of the third production term. But because (?v/~x) changes sign, this term averages to zero in the streamwise direction. The second term exhibited the same characteristics but was less influential due to the fact that @u/?y) was relatively weak in the Q2 events. This term was, per definition, strong in the PQ2 events. The characteristic features of the other production terms were fairly independent of the detection method, except Vf(~?u/Ox) which was of relatively small importance. Ensemble averages of the terms involving pressure (which are the same as those that occur in the time-averaged sense) are plotted in Fig. 9. Figs. 9a and 9b display the pressure-strain correlations. In the case of -(p~?u/~yl, relatively strong contributions in the near-wall layer are seen to be associated with the Q2 events at the centreline. This is partly because the pressure can be regarded as a global variable (since it can be derived from an integral of its Poisson equation) and therefore correlations prevail over larger length scales. In addition, (u) exhibited large correlation lengths, as can be inferred from Fig. 6a. In the time-averaged sense, -(p~?u/~?y) was the most important contribution in the near-wall region. In the central region, however, this term became less influental because (p) changes sign around x = 0 and the term will thus be reduced when averaged in the x-direction. The
76
K.H. Bech, H.I. Andersson / Fluid Dynamics Research 18 (1996) 6 5 - 7 9
a)
t
1.0 [ 0.0
.................... ..:.,
1 f:: ':"':'
'"' "" '::'::'
b)
:" " :":
-1.0
c)
-1.0 .... ::::::::::::::::::::::::: . . . . . . . 1.0 / ,. ,---,,,--:::,-.-.-,,.
:::'--?. . . . . :
d)
1
-1.o -4.0-3.0-2.0-1.o o.o 1.o 2.0 3.0 4.0 Fig. 8. Conditionally averaged pressure terms obtained from Q2-detection. (a) - ( p Ou/~y); (b) - ( p Ov/~x); (c) (u ~ p/~y); (d) (v 3p/~x);. Contour increment 0.05. Non-dimensionalized as in Fig. 3.
-(p~v/Ox) was the most important pressure term in the central region. It acted as a sink for -(uv) because both (p) and @v/~x) changed sign around x = 0. The velocity-pressure gradient cor-
term
relations in Figs. 9c and 9d appeared to be very similar to the corresponding pressure-strain correlations, except that the near-wall effects were weaker. This is in accordance with the fact that a contraction of the time-averaged pressure terms, cf. Eq. (4), reduces the complexity of the dynamics of -Fg. By comparing Figs. 8c and 9c, an interesting feature arises. There seems to be a strong negative correlation between the production t e r m uZ(Ov/Ox) and (u~p/Oy). These two terms seem to cancel each other out locally because the number of contours were similar. An analogous observation is valid for the terms (v2)dU/dy and (vOp/~x). This pairing of the major source and sink terms is by no means obvious.
4. Final discussion and concluding remarks The present result for the mean shear parameter, S = 0.22, compared favourably with the values 0.21 and 0.20 which Tillmark and Alfredsson (1993) deduced from the measurements of Aydin and Leutheusser (1991) and their own experimental results, respectively. These investigations were carried out at approximately the same Reynolds number. At the somewhat higher Re = 3000, the mean velocity profile obtained in the DNS of Lee and Kim (1991) showed that S < 0.20, although the authors claimed that their value of S was "about 30% of the value in the equivalent laminar flow with a linear velocity profile". It is interesting to recall that the pioneering experiments of Reichardt (1956) showed that the dimensionless slope in the central region of the Couette flow decreased with Re, and that Busse (1970), by means of a variational method, found the lower bound for infinite Re to be 0.25. The present finding and the experimental data collected by Tillmark and Alfredsson (1993) obviously do not support Busse's conclusion.
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79
77
If one assumes -V~ to be constant in a certain region, the viscous shear stress must also remain constant according to Eq. (1). This moreover implies that the production rate of turbulent kinetic energy, Pk, should be constant in that region. None of these assumptions were in accordance with the present results for the core region of the Couette flow. The viscous shear stress was not negligible and exhibited a relatively large variation, while the relative variation of the turbulent shear stress was modest. The variation of the mean velocity profile, as derived by von Karman (1937), showed reasonable agreement with the present results. Von Karman adopted the local equilibrium hypothesis, which turned out to be a reasonable first-order approximation because turbulent diffusion of kinetic energy was small. However, the central region of the Couette flow at low Reynolds numbers cannot be considered as truly homogeneous because the mean shear and some one-point statistics vary substantially. The Couette flow simulation (DNS) by Lee and Kim (1991) revealed a persistent vortical seeondary flow pattern in the form of counter-rotating streamwise vortices which had not been observed experimentally. The present simulation did not support the possible existence of streamwise vortices in the central region of turbulent Couette flow. The instantaneous structure of the flow field resembled large, elongated regions of altemating u fluctuations in combination with localized peaks in i v , and the ensemble averages obtained with Q2 detection exhibited simple spatial structures with length scales typical of the turbulence statistics. In this context, it may be interesting to recall that the DNS of homogeneous shear flow by Lee et al. (1990) indicated that a high shear rate alone is sufficient for generation of streaky structures, and that the presence of a solid boundary is not necessary. However, the present value 5.7 of the dimensionless shear rate parameter 2(dU/dy)k/e at the centreline of the Couette flow classifies the core region as a low-shear-rate flow in which streak-like structures should not be expected. There was no one-to-one correspondence between Q2 and PQ2 events and none of them could be considered as a subset of the others. In some of the PQ2 events, generation and destruction of the negative uv-correlation probably occurred simultaneously. Some of the Q2 events could be caused by other localized production terms, see Eq. (5), and by local positive correlations between velocity and pressure gradient. These differences may partly be due to the different detection methods and the adopted threshold values. The ensemble-averaged generation and annihilation terms were significantly correlated and with opposite signs, i.e. production and velocity-pressure gradient destruction of - u v occurred simultaneously. Insofar as second-moment turbulence modelling is concerned, the traditional way of handling correlations involving pressure is a decomposition into a pressure diffusion dlP2 and a pressure-strain correlation q~ij, i.e. in accordance with Eq. (3). The classical physical interpretation of the latter correlation, namely that the traceless tensor ~bij is responsible for an intercomponent energy transfer, involves only its diagonal components; cf. the pictorial arguments of Hinze (1975). The off-diagonal components have, on the other hand, traditionally been treated as the by-product of the decomposition and subsequent modelling of the pressure terms. By means of a coordinate transformation so that the coordinate axes became the principal axes of the Reynolds stresses, Hinze demonstrated that ~b~2 and -~-~ have opposite signs. Although Hinze's conclusion is in accordance with the present results (cf. Fig. 3), we do not accept his reasoning. By considering the shear stress, the traditional decomposition of the pressure terms does not necessarily ease the search for a physical interpretation. To this end the contraction in Eq. (4) should be adopted, and it has already been shown that vDp/3x represents the major contribution to the velocity-pressure gradient term H12 at the centreline.
78
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79
Let us now consider the Q2 events depicted in Fig. 6. These events represent lumps of fluid moving in the positive y-direction (v > 0) with a streamwise velocity lower than their surroundings (u < 0). Thus, from the viewpoint of the faster moving surrounding fluid, the slowly propagating lumps are felt as obstacles. Since the excess velocity of the surrounding fluid is in the positive x-direction, a high-pressure zone should be expected on the upstream side of the lumps and a lowpressure region on the downstream side, i.e. analogous to the well-known pressure distribution around a solid sphere in a parallel stream. This conceptual model is supported by the observed pressure field in Fig. 6d. Moreover, the lump of fluid with velocity v > 0 should therefore be associated with a negative value of Op/~x so that vdp/~x < 0. This simple physical interpretation is thus in full accordance with the averaged velocity-pressure gradient term vOp/~x in Fig. 4. The Q2 events will, of course, also contribute favourably to -uv. The strong events considered here, moving in a background mean shear dU/dy, give rise to a significant positive correlation (v2)dU/dy. Thus, the Q2 events both create and destroy the correlation - u v . Both ensemble averages and time-averaged statistics showed that the two correlations, (vZ)dU/dy and (vOp/~x), were of crucial importance for the dynamics of - u v in the central region. Because the time-averaged results were partly reproduced by the ensemble averages, it can be concluded that the conditional sampling procedures were able to detect events representative for the structure of the flow. This implies that extremely strong events capture important features of the dynamics of - u v . In these events (v2) exhibited a peak value near that of the streamwise fluctuations (u2). In pressure-driven Poiseuille flow, however, such events seem less likely to occur because vz there exhibits a local minimum at the centreline. In the Poiseuille flow, near-wall structures, such as those captured by the VISA-technique, may be responsible for the major part of the turbulent shear-stress generation. Very similar structures were found in the near-wall region of the shear-driven Couette flow (see Bech et al., 1995). In the central region, however, different processes, some of which are described above, were responsible for generating and maintaining the high and almost uniform value of the turbulent shear stress in the Couette flow.
Acknowledgements This work has received support from The Norwegian Supercomputing Committee (TRU) through a grant of computing time. We are thankful to Professor P.-A. Krogstad at the Norwegian University of Science and Technology in Trondheim for commenting upon the results in Section 3.2. Dr. N. Tillmark at the Royal Institute of Technology in Stockholm made some valuable comments upon the two-point correlations. We are also grateful to the referees for their comments.
References Afzal, N. (1993) Asymptotic analysis of turbulent Couette flow, Fluid Dyn. Res. 12, 163. Andersson, H.I., K.H. Bech, and R. Kristoffersen(1992) On diffusionof turbulent energy in plane Couette flow, Proc. R. Soc. Lond. A 438, 477. Andersson, H.I. and B.A. Pettersson (1994) Modelling plane turbulent Couette flow, 25th AIAA Fluid Dynamics Conf., Paper 94-2342; Int. J. Heat Fluid Flow 15, 447. Aydin, E.M. and H.J. Leutheusser(1991) Plane-Couetteflow between smooth and rough walls, Exp. Fluids 11, 302.
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79
Bech, K.H. and H.I. Andersson (1994) Very-large-scale structures in DNS, in: P.R. Voke, L. Kleiser and J.-P. Chollet, eds., Direct and Large-Eddy Simulation I (Kluwer, Dordrecht) p. 13. Beth, K.H., R. Kristoffersen and H.I. Andersson (1993) Inner-layer velocity statistics in plane Couette flow, Int. Conf. Near-Wall Turbulent Flows, Tempe, Arizona, p. 317. Bech, K.H., N. Tillmark, P.H. Alfredsson and H.I. Andersson (1995) An investigation of turbulent plane Couette flow at low Reynolds numbers, J. Fluid Mech. 286, 291. Busse, F.H. (1970) Bounds for turbulent shear flows, J. Fluid Mech. 41,219. El Telbany, M.M.M. and A.J. Reynolds (1982) The structure of turbulent plane Couette flow, ASME J. Fluids Eng. 104, 367. Gavrilakis, S., H.M. Ysai, P.R. Voke and D.C. Leslie (1986) Large-eddy simulation of low Reynolds number channel flow by spectral and finite difference methods, Notes Numer. Fluid Mech. 15, 105. Gersten, K. (1985) The turbulent Couette flow from asymptotic theory view point, in: G.E.A. Meier and F. Oberrneier, eds., Lecture Notes in Physics, Flow of Real Fluids (Springer, Berlin) p. 219. Groth, J. (1991) Description of the pressure effects in the Reynolds stress transport equations, Phys. Fluids A 3, 2276. Hinze, J.O. (1975) Turbulence, 2nd ed. (McGraw-Hill, New York) p. 325. Karman, T. von (1937) The fundamentals of the statistical theory of turbulence, J. Aeronaut. Sci. 4, 131. Kim, J., P. Moin and R. Moser (1987) Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech. 177, 133. Kristoffersen, R., K.H. Bech and H.I. Andersson (1993) Numerical study of turbulent plane Couette flow at low Reynolds number, Appl. Sci. Res. 51, 337. Lee, M.J. and J. Kim (1991) The structure of turbulence in a simulated plane Couette flow, Proc. 8th Symp. on Turbulent Shear Flows, Munich. Lee, M.J., J. Kim, and P. Moin (1990) Structure of turbulence at high shear rate, J. Fluid Mech. 216, 561. Lu, S.S. and W.W. Willmarth (1973) Measurements of the structure of the Reynolds stress in a turbulent boundary layer, J. Fluid Mech. 60, 481. Lund, K.O. and W.B. Bush (1980) Asymptotic analysis of plane turbulent Couette-Poiseuille flows, J. Fluid Mech. 96, 81. Mansour, N.N., J. Kim and P. Moin (1988) Reynolds-stress and dissipation-rate budgets in a turbulent channel flow, J. Fluid Mech. 194, 15. Missimer, J.R. and L.C. Thomas (1983) Analysis of transitional and fully turbulent plane Couette flow, J. Lubr. Tech. ASME 105, 364. Papavassiliou, D.V. (1993) Direct numerical simulation of plane Couette flow, MS Thesis, Univ. of Illinois, UrbanaChampaign. Reichardt, H. (1956) Ober die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couettestr6mung, Z. Angew. Math. Mech. 36, 26. Reichardt, H. (1959) Gesetzmfissigkeiten der geradlinigen turbulenten Couettestr6mung, Max-Planck-Institut ffir Str6mungsforschung. Schneider, W., R. Eder and J. Sehmidt (1990) Turbulent Couette flow: asymptotic vs. experimental data, in: Nayfeh et al,, eds., Proc. 3rd Int. Congress on Fluid Mechanics, IV, 1593. Tillmark, N. and P.H. Alfredsson (1993) Turbulence in plane Couette flow, Appl. Sci. Res. 51, 237.
Fluid Dynamics Research 18 (1996) 65-79
Structure of Reynolds shear stress in the central region of plane Couette flow K n u t H. B e c h , H e l g e I. A n d e r s s o n *
Department of Applied Mechanics, Thermodynamics and Fluid Dynamics, Norwegian University' of Science and Technoloqy, N-7034 Trondheim, Norway Received 27 March 1995; revised 22 December 1995; accepted 29 January 1996
Abstract The structures and mechanisms for maintaining a high Reynolds shear stress throughout the core region of plane turbulent Couette flow were examined by means of databases originating from a direct numerical simulation. At the relatively low Reynolds number considered, the mean shear rate and some one-point statistics exhibited a non-negligible variation in the core region, which conflicted with the postulated homogeneity. The slope of the mean velocity profile was below a theoretically established lower bound for the limit of infinite Re. Analysis of the time-averaged and structural information obtained by conditional sampling showed that mean shear generation and velocity-pressure gradient correlations played a crucial role in the generation and annihilation of the Reynolds shear stress. It was moreover observed that strong, very localized velocity fluctuations in the wall-normal direction were essential in both processes. A simple conceptual model was proposed to explain the physical significance of the pressure field associated with quadrant 2 events.
I. Introduction In spite o f being a classical problem in fluid mechanics, the shear-driven motion o f an incompressible fluid bounded by two walls in relative motion is o f practical interest in a variety o f contexts, ranging from lubrication technology to thin-film coating. The simplest flow o f this kind is the plane Couette flow, i.e. the flow between parallel planes in which no average pressure gradient exists in the flow direction. The fluid flow is maintained solely by the uniform motion o f the planes, and the Reynolds number (based on the relative wall speed) determines whether the flow is in the laminar or turbulent regime. The fully developed turbulent plane Couette flow is a paradigm o f turbulent shear flows, in which the total shear stress dU
=
*Corresponding author.
0169-5983/96/$15.00 @ 1996 The Japan Society of Fluid Mechanics Incorporated and Elsevier Science B.V. All rights reserved PIlSO 1 6 9 - 5 9 8 3 ( 9 6 ) 0 0 0 0 8 1
(1)
66
K.H. Bech, H.I. Andersson/Fluid Dynamics Research I8 (1996) 65 79
is constant (and equal to its wall value zw) throughout the entire flow ~. Moreover, since the mean velocity vector V = ( U ( y ) , 0 , 0 ) is aligned with the walls, the motion is homogeneous in planes parallel to the bounding walls, i.e. the mean value of any combination of the turbulent fluctuations becomes a function of the single coordinate (y) measuring the distance from the channel centre. The fully developed Couette flow is therefore particularly amenable to theoretical analysis (von Karman, 1937; Busse, 1970; Lund and Bush, 1980; Missimer and Thomas, 1983; Gersten, 1985; Schneider et al., 1990; Afzal, 1993) and phenomenological modelling (see, e.g. Andersson and Pettersson (1994) for references to one-point closure modelling). In spite of its fundamental nature, however, the plane Couette flow is difficult to realize in the laboratory, and the number of reliable experimental studies are accordingly scarce. The pioneering laboratory investigation is that of Reichardt (1956, 1959) whereas detailed turbulence statistics were provided by E1 Telbany and Reynolds (1982) and more recently by Aydin and Leutheusser (1991) and Tillmark and Alfredsson (1993). Further statistics and structural information have been made available by direct numerical simulations (DNSs) (Lee and Kim, 1991; Papavassiliou, 1993; Kristoffersen et al., 1993) and an elaborate comparison between near-wall data from both numerical and physical experiments was made by Bech et al. (1995). The aforementioned studies have revealed some prominent features of turbulent Couette flow, namely that the near-wall regions are practically the same as in pressure-driven Poiseuille flow and flat-plate boundary layers, whereas the central region closely resembles that of homogeneous turbulent shear flow. The latter part of this statement is deliberately vague since some conflicting hypotheses still prevail in the literature. While the core region is undoubtedly anisotropic due to its non-vanishing Reynolds shear stress, it still remains an open question whether the core flow is homogeneous in the wall-normal direction or not. Under the assumption of constant shear stress -puT, yon Karman (1937) postulated the existence of "homologous" turbulence, i.e. "the correlation coefficients are independent of the location of the point considered" (although they are not invariant as far as rotation and reflection are concerned). Moreover, based on the assumed constancy of -puT, von Karman derived a formula for the mean shear rate, viz., dU/dy cx 1/cos(~y +/~), where and fl are integration constants. According to Eq. (1), however, a uniform mean shear rate is inevitably implied by the assumed constancy of the Reynolds shear stress. The variation of the mean velocity deduced by von Karman therefore conflicts with the underlying assumption of "homologous" turbulence. Later theoretical studies have also led to conflicting conclusions. By means of a variational method, Busse (1970) found an extremalizing solution for plane Couette flow, which exhibited a constant mean shear rate. A comparison with the experimental data of Reichardt (1959) showed that the measurements reflected the theoretically established lower bound Uw/4h on the mean shear rate dU/dy in the interior of the plane Couette flow. On the contrary, Lund and Bush (1980) showed through the use of matched asymptotic expansions that the mean velocity gradient becomes vanishingly small in the core region as the Reynolds number tends to infinity. This contrasts with the measurements by Aydin and Leutheusser (1991), which showed that the logarithmic part of the velocity profiles extended right to the centre of their Couette channel.
l ln the following, U is the mean (streamwise) velocity and u, v, w are the fluctuating velocity components in the streamwise (x), wall-normal (y) and spanwise (z) directions, p and p denote the constant fluid viscosity and density, respectively.
K.H. Bech, H.I. Andersson/ Fluid Dynamics Research I8 (1996) 65 -79
67
The present study focuses on the nearly homogeneous core region of the turbulent Couette flow. The non-vanishing turbulence production makes this part of the flow fundamentally different from that of plane Poiseuille flow, in which not only the turbulence production but also the Reynolds shear stress vanish at the centre plane. The objective of the investigation is therefore twofold, namely: (a) to explore the dynamics of the cross-correlation ~-~ (i.e. the turbulent shear stress), and (b) to examine the postulated homogeneity of the central region. This will be accomplished by means of statistical (long-time averages) and structural (conditional sampling) information deduced from the direct simulation reported by Bech et al. (1995).
2. The numerically generated databases and sampling techniques The present DNS was carried out utilizing the computer code ECCLES by Gavrilakis et al. (1986). The incompressible momentum equations are discretized by second-order accurate central differences. The Poisson equation for the pressure is Fourier-transformed with respect to the streamwise and spanwise homogeneous directions, and the resulting equations are solved directly for each time step. The time advancement is implemented through the Adams-Bashforth scheme. No-slip boundary conditions are used for the velocity components at the walls, while periodicity is assumed in the xand z-direction. Consider plane Couette flow where the parallel walls move in opposite directions. The lower wall located at y = - h propagates with velocity -Uw and the upper wall at y --- h moves with Uw in the streamwise (x) direction. The bulk Reynolds number, Re = Uwh/v, was 1300, where v = l~/P is the kinematic viscosity. The Reynolds number based on friction velocity (u, = x/~w/P) became Re~ = u~h/v = 82.2. The size of the computational box was 10nh x 2h x 4nh in the streamwise, wall-normal and spanwise directions, respectively. The corresponding number of grid points was 256 x 70 x 256. The computational mesh was non-uniform in the y-direction, the grid point next to the walls being at y+ ~- 0.35, whereas the largest spacing y+ -~ 3.9 was at the centreline. For comparison, the equally distributed grid points in the homogeneous xz plane yielded Ax + -~ 10.1 and Az + --~ 4.0. The time step was kept constant and equal to O.03v/u2~. This simulation supercedes the DNS reported by Kristoffersen et al. (1993) with respect to grid resolution and box size. A comparison between the two simulations was made by Bech and Andersson (1994), while a more complete description of the new simulation was given by Bech et al. (1995), in which detailed comparisons between computed and measured turbulence statistics are provided. The statistical results presented here will be taken from two different databases. The time-averaged statistics were sampled during 16.4h/u~ and constitute the first database (the statistics database). The second, which was used for conditional sampling and calculation of auxiliary terms, consisted of 12 fields of velocity and pressure, i.e. a complete description of the flow in the computational box at 12 different time levels. The time separation between the fields was O.15h/u~. The latter data will be referred to as the field database. The total sampling time for the field database was about 1.65h/u~, which should be adequate. Because of the relative short time interval between the fields, one event could have been detected several times during its evolution. This is also the case for the time-averaged statistics, and was not considered as a problem as long as a sufficient number of different events was detected.
68
K.H. Bech, H.L Andersson/Fluid Dynamics Research 18 (1996) 65-79
In Section 3.2, results on conditional sampling will be presented. Three different detection schemes were tested in the plane y --- 0, i.e. the centre plane. The first scheme, which will not be reported here, detected regions with large wall-normal momentum. The second was the quadrant splitting detection due to Lu and Willmarth (1973), where it was demanded that u and v should be in a certain quadrant in the uv-plane and that uv/V~ should exceed some threshold level tQ2. Only events in the second quadrant (Q2) were considered here. The last scheme applied detected events with high instantaneous production of -uv, i.e. V2(Cgu/Oy) p
v2(dU/dy)
~>tpQ2 and u < 0,v > 0,
(2)
where tpQ2 is some threshold level. Note that only quadrant 2 events were considered, hence the abbreviation PQ2 (production by Q2-events) will be applied for the scheme described by Eq. (2).
3. Results In this section, results on the statistical properties as well as the dynamics of the turbulent shear stress will be presented. The first subsection deals with time-averaged statistics, the second is concerned with conditional sampling and ensemble averages. The time-averaged quantities provide definite information concerning -~-0. The ensemble averages yield information on the kinematics of events involved in the generation and destruction of the negative correlation between u and v. In this section, and throughout the rest of the paper, all variables have been non-dimensionalized by the channel half-width h and the friction velocity u,. In order to keep the notation as simple as possible, the non-dimensional variables are not distinguished from the corresponding dimensional variables in the first part of this paper. It might be argued that u, is not a typical velocity in the central region. An alternative choice in a region which is close to local equilibrium is q / = (eh) 1/3, where e is the dissipation rate of turbulent kinetic energy. Assuming local equilibrium, u,/#i is approximately proportional to RG-1/3 and thus near order unity for small Reynolds numbers. In order to rescale the present results, the factor u~/~ = 0.63 (for y = 0) should be applied.
3.1. Time-averaged dynamics of the turbulent shear stress The results presented in this subsection will be taken from the statistics database. The overbar denotes averaging in homogeneous m-planes and time. Note that data from the two half-channels were averaged and that the wall is at y = - 1 and the centreline at y = 0. The first figure visualizes the terms in Eq. (1). The curves were found to be smooth and the sum of the turbulent and viscous shear stresses was constant within 0.2% from wall to channel centre. This constancy indicates statistically converged results. The stress curves intersect at y = -0.872 or y+ = 10.5 where the maximum production of turbulent kinetic energy (Pk = -V~dU/dy) occurs. The partition among the viscous and turbulent contributions to the total shear stress varied substantially across the flow. At the centreline, for example, the viscous stress Re~ l dU/dy and the turbulent stress - ~ contributed 4.5% and 95.5%, respectively. Fig. la shows that - ~ remained practically constant (to within 1%) throughout the central region, i.e. y > -0.4, thereby supporting
K.H. Bech, H.I. Andersson/FluidDynamicsResearch18 (1996) 65-79
69
a)
1.2
,
,
~
,
,
, - ~ - - ~ ,
,
b) 0.l
,
....
, ....
1
o
0.075 0.8
o o
\
o o \\
0.6
0.05
0.4 0.025 0.2 0
/,
,
,
i
i
-0.8
-1
-0.6
i
-0.4
I
0
-0.2
0
-
~ t
-0.5
,
,
,
-0.25
,
0
Y Fig. 1. (a) Viscous and turbulent shear stress from wall to centreline. (b) Viscous shear stress in the central region compared with various theories. ( - - ) -~-f; ( - - - ) (- - -) -riO+ (r-l) -fi-f,,~; (o) logarithmic profile, 1/(~c(1 + y)Re,), ~c = 0.284; ( x ) von Karman theory, cos[roy/2]), k = 0.446. k and x was found from at the centreline.
Re~-]dU/dy; (~z/2)/(kRe,
dU/dy
Re~-]dU/dy;
a) 8
~
4 I ...
,
,
i
b) i
. . . .
,
"
i
"""
-... :,..~',-~::~..::~,
0
~~~'..~-.-.--4
- i
-t
l" /
-8
i~'
-8
-4
0 U
i
i
4
i
i
i
8
-8
-4
0
4
8
U
Fig. 2. Quadrant plots in the uv-plane at (a) y = - 0 . 5 and (b) y=0. Broken lines denote fourth grid point at one instant are shown•
uv I=
10. Values at every
the frequently assumed constancy of-fi-~. The mean shear parameter S = d(U/U~,)/d(y/h) was 0.22 at the centreline. A close-up of the viscous shear stress Re[ldU/dy away from the walls is shown in Fig. lb. The corresponding variation deduced from a logarithmic velocity profile, i.e. dU/dy = 1/to(1 + y), as well as the variation derived from the analytical formula proposed by von Karman (1937), i.e. dU/dy = (½n)/kcos(½nY), have also been included. The constants ~c and k were obtained from the mean shear at the centreline where dU/dy = SRe/Re~ = 3.52. It is readily observed that the solution deduced from the logarithmic profile, as advocated by Aydin and Leutheusser (1991), seriously exaggerates the variation of the mean shear, whereas von Karman's proposal compares favourably with the DNS data in the core region. Quadrant plots of (u, v) are presented in Fig. 2. Extremely strong negative correlations between u and v were found to exist. The fluctuations were of approximately the same strength at the centreline
70
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79
and half way towards the wall. In the turbulent boundary layer, fluctuations with [ uv I exceeding 4.5 were characterized as large by Lu and Willmarth (1973), while the broken lines in Fig. 2 represent I uv ]= 10. The probability density distribution of the fluctuating Reynolds shear stress was very different from the Gaussian with a skewness factor of about 4.5 and flatness (kurtosis) slightly exceeding 20. The latter is consistent with the flatness factor at y+ ~- 80 in the Poiseuille flow simulation of Kim et al. (1987), while their skewness factor was about 3. This suggests that uv events are equally intermittent in Poiseuille and Couette flow, whereas the probability density function is more asymmetric in the latter flow in which structures with significant turbulent shear stress contribute to the mean 2. The dynamics of the turbulent shear stress -VO is described by the following non-dimensional transport equation, e.g. Hinze (1975):
8t8(_fi_~)~dU=v ~y + -dyUVd'+~"2~ypud(~?u_ P~y + P~xOV)-v~--~2y2fi-O+2v~ avcgxk = Pl2 + d/12+ dP2 + ~b,2 + d~2 - ~,2,
(3)
where the symbolic notation for the various terms has been introduced for convenience. Because -V~ was of primary concern in the present study, the signs in Eq. (3) have been inverted as compared to Hinze's dynamic equation for the correlation ~-~. In the statistically steady state considered here, ~ ( - V ~ ) / ~ t = 0 and the delicate balance between the different terms on the right-hand side of Eq. (3) is presented in Fig. 3a. The observed variations of the various terms in the near-wall region are qualitatively the same as in the plane Poiseuille flow investigated by Mansour et al. (1988). As in Poiseuille flow, the complexity of the -~T-budget is reduced in the central region, both with respect to the number of significant terms and their variation. The dominating contributions were the production P~2 and the pressure-strain correlations ~bl2, which both exhibited a modest monotonically reduction towards the centreline. The viscous diffusion d~2 was practically negligible for y > -0.5, while the viscous dissipation rate e12 attained an approximately constant level El2 < 0.01 in the central 40% of the channel. Likewise, the turbulent diffusion due to velocity (dtl:) and pressure (dP2) fluctuations remained constant in the core region (y > -0.4). The constancy of d~e and dr2 revealed that the covariances uv 2 and pu varied linearly with y in the core region of the Couette flOW.
It is noteworthy that d~2 exhibited the same behaviour as the diffusion of turbulent kinetic energy, to which - d ( u Z v ) / d y was the dominating contribution. The strong positive correlation between -uZv and uv: is obviously due to the negative correlation between u and v away from the walls. This moreover suggests that there was no fundamental difference in the turbulent transport in the ydirection of the shear stress -k--~ compared to the normal stress u 2. The dissipation was found to be a relatively modest sink term except in the vicinity of the walls (~12/P12 = 0.125 at y = 0) compared to its dominating role in the budget for the turbulent kinetic energy where e/Pk = 1.175 at y = 0. This could be expected since e~2 consists of relatively weak cross correlations between velocity gradients as opposed to e = v(OujSxk)(SuJOxk). 2Here, it should be recalled that the skewness of uv inevitably goes through zero at the symmetryline (y+ = 180) in the Poiseuille flow simulation.
K, H. Bech, H. L Andersson / Fluid Dynamics Research 18 (1996) 65-79
71
a)
b _ l t l l j r
0.1
0.05
0
0
-0.1
-0.05 •
-1
-0.8
~
,
-0.6
,
I
-0.4
J
,
,
[
,
-0.2
y
,
I
J
0
-0.5
t
r
I
i
]
I
-0.25
I
i
r
i
I
0
y
Fig. 3. Terms in the transport equation for - ~ f . (a) Conventional splitting, all terms shown from wall to centreline. (b): Close-up of central region, pressure terms added. (- - -) dt12; (--) d~'2; ( - - - - - - ) de2; ( - - - ) -~12; ( x ) ~bl2; (o) P12; ( . ) dP2 + q~12; ( . . . . ) sum of terms (scarcely visible). Non-dimensionalized with u~/v.
The most important sink term in Eq. (3) was the pressure-strain term ~D12. Unlike the diagonal elements of ~bij, which redistribute energy among the normal Reynolds stress components, the offdiagonal terms (i # j ) are not redistributive. This suggests that the present decomposition of the pressure-velocity interactions, as advocated recently by Groth (1991), does not necessarily ease a sound physical interpretation. Here, it was found convenient to contract terms involving pressure fluctuations to a velocity-pressure gradient term
I-I,2 -= ~12 + dP2 -~ U~y + V~x.
(4)
This further simplifies the dynamics of - ~ , as shown in the close-up of the central region in Fig. 3b. A more detailed examination of the different pressure terms can be made from Fig. 4, in which the individual contributions to ~/)12 and H12 have been plotted. None of the terms, which all acted as sinks in Eq. (3), turned out to be negligible. The major contribution to H~2 in the near-wall region was uOp/~y, but this term gradually decreased towards the centre where v~p/t3x dominated. It is moreover noteworthy that the variation of vOp/t3x coincides with that of -pOv/Ox. This simply reflects the fact that the streamwise pressure diffusion d(~-~)/dx should vanish in fully developed flows. The equivalence of the independently computed terms vt3p/Ox and -p~3v/~x in Fig. 4 therefore supports the adequacy of the present sampling time.
3.2. Structures obtainedfrom conditional samplin9 Ensemble averages obtained from conditional sampling may give an impression of the length scales and structure-characterizing events that are important for the dynamics of the turbulent shear stress.
72
K.H. Bech, H.L Andersson/Fluid Dynamics Research 18 (1996) 65-79
0.02
...
~
. ~
• ~
0
. o
x
o
o
•
o
o
~
°
o
-0.02 -0.04 -0.06 -0.08 -1
-0.8
-0.6
-0.4
-0.2
0
Y Fig. 3. Pressure terms in the transport equation for - E l . (o) - p 8u/Oy; (x) - p Ov/~3x; ( - - ) u ~?p/~3y; (- - -) v c3p/dx. Non-dimensionalized as in Fig. 3. Bech et al. (1995) showed that near-wall internal shear layers detected by the VISA 3 technique contributed substantially to -V~. These events were, however, confined to the near-wall region and not associated with the high level of the Reynolds shear stress in the central region. The mean shear is substantially higher in the near-wall region than in the central region, implying that the turbulent structure near the centreline is different from the more universal near-wall structure. In fact, VISA-detection was also conducted in the central region, but the detected events were weak and very scarce. So detections around the centreline should preferably be made with a method that triggers directly on - u v rather than on some specific structure. The detection algorithms, as described in Section 2, were applied to the field database. For the purpose of Q2 detections, a threshold value to2 --- 10 was adopted. From Fig. 2 it can be inferred that relatively few events with extreme negative correlations between u and v were selected. The events detected at one instant in the xz-plane at y = 0 were plotted in Fig. 5a. The number of islands (detections) in the plane was 41 4. In Fig. 5b, the events detected with the PQ2 scheme at the same instant can be seen, the number of PQ2 islands being 294. The threshold level was tpQ2 = 20, and therefore only regions with extremely strong instantaneous production were taken into account. The rationale for examining extreme events was that a spatial structure would not be blurred by the ensemble averaging. Some of the events detected simultaneously by both the Q2 scheme and the PQ2 scheme were marked with arrows. There was no one-to-one correspondence between strong negative correlation between u and v and strong instantaneous production of the correlation. The areas detected with the Q2 algorithm were generally larger than the PQ2 islands, probably because the latter scheme detects partly on gradients which tend to have shorter length scales than the variables themselves. The ensembles were generated by sampling flow variables in volumes centred around the maximum of each event. That is, u, v, w and p in regions of size 7.24 x 2 x 2.41 centred around the peak of each island in Fig. 5 were averaged. This procedure was followed for each of the 12 fields in the database, and one ensemble was generated by each of the detection schemes. 3Variable interval space averaging. 4The number is an average over the field database.
K.H. Bech, H.I. Andersson / Fluid Dynamics Research 18 (1996) 65 -79
73
a) 10o
0
,
b)
0
10
-~,,,
o °',k,
" "
o~
0
.
0
.
.
.
,
.
°
*'
.
10
.
,
20
~.
.
•
.
r
30
X
Fig. 4. Events detected by (a) the Q2-scheme and (b) the PQ2-scheme. The contour levels correspond to the detection threshold levels. Some of the events detected by both schemes are pointed out by arrows. Averages of the ensemble (denoted by angular brackets) obtained with Q2 detection can be studied in Fig. 6. The diagrams, showing a cross-section of the ensemble averages at z = 0, give some information on the length scales of the extreme events considered. The corresponding ensemble averages in the xz-plane (not shown here) gave length scales that compared well (within 5%) with length scales obtained from the spanwise two-point correlations for u and v derived from the statistics database. A further observation from the two-point correlations was that v had the same length scales in the homogeneous directions x and z. Because of the observed agreement between time-averages and ensemble averages with respect to length scales, it was assumed that the conditional sampling procedure captured the essential features of the events and not only a fraction of them. It is interesting to note that (u) in Fig. 6a extends out to the walls, i.e. events at y = 0 have correlation lengths e( h. The outer contours are sligthly deformed by the mean shear. Only topographically simple velocity structures can be seen, i.e. {u) and {v) exhibit a single minimum and maximum, respectively, thus yielding a maximum in - { u v - V~) in Fig. 6c. It is evident that the Q2 scheme detects events with a wide span of length scales and that the ensemble average will not provide elaborated structures. Besides, averages based on one-point detection schemes cannot reproduce spatial asymmetries of the individual events. The pressure contours in Fig. 6d did, however, exhibit an interesting asymmetry, assumingly because only Q2 (and not Q4) events were considered. The Q2 events may be thought of as obstructions in the mean flow, i.e. lumps of fluid travelling with lower speed than the surrounding fluid and therefore giving rise to high-pressure upstream and low-pressure downstream of the lump. This pictorial model is partly confirmed by the pressure contours and also by plots of ensemble averages in the xzplane. Corresponding ensemble averages obtained with the PQ2 scheme are displayed in Fig. 7. The structures detected by the two different schemes differed mostly with respect to the streamwise fluctuating velocity. The PQ2 ensemble was more asymmetric with respect to y = 0 because strong instantaneous gradients ~?u/~?y were a dominant feature of the detected events. The higher number of
74
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79 1.0
0.0
-I.0 1.0
t .
:::::::::::::::::::::::::::::::::
......
a)
............ ~.";;::~-:.!:~.--%;...i-:-~:U". . . . . . . . ' ,.-" . ....... ~,,:':!:i:!;!?(~.!~:~,:,", "," ", . . . . . - - . - . . , . ,.:.,~,,,.'.f~,,.,.;.,,,:,: . . . :
,o I
-1.0 1.0
."
,'""..._
b)
. c)
o.o] -1.0
:::,' 0.0 -1.0
"::-~ .... '~
d)
.,'
........
-4.0-3.0-2.0-1.0
0.0 1.0 2.0 3.0 4.0
Fig. 5. Ensemble averages obtained with Q2-detection. (a) (u), contour increment 0.25; (b) (v), contour increment 0.25; (c) - ( u v - ~-6), contour increment 1.0; (d) (p), contour increment 0.5. Broken lines correspond to negative values. Non-dimensionalized with u~. 1.0
o.oI
a)
.......... , ....:-?,~:~¢~/~.:.-:.................. .. ......... ::::: I~::<" ....... :;;::"
I.O 0.01
b) ~
o
-i.0 1.0
c)
-1.0 1.0
d)
o.oI
1o"0 -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0
Fig. 6. Ensemble averages obtained with PQ2-detection. Caption as in Fig. 6.
pressure contours obtained with the PQ2-detection was probably due to the fact that the events exhibited strong u-velocity gradients, and these gradients contribute to the pressure distribution through the integrated Poisson equation for the pressure. By applying ensemble averaging instead of time averaging, a modified version of Eq. (3) can be obtained. The number of terms originating from the non-linear terms of the Navier-Stokes equations
K.H. Bech, H.I. Andersson/FluidDynamics Research 18 (1996) 65-79 1.0 0.0 -1.0 1.0 0.0
] ]
75
a)
b) Q?.
-1.0 " ".... :: ,;2 {:" :i.--..:"-
'°i
c)
0.0
-1.0 '~ " ,', ,;G" ':---,--,-;--::..: :/-. -4.0 -3.0 -2.0 -1.0 0.0 1.0 2.0 3.0 4.0 X
Fig. 7. Conditionally averaged production terms obtained from Q2-detection. (a) (v2)dU/dy; (b) ~(Ou/~?y};(c) ~@v/~?x}. Contour increment 0.05. Non-dimensionalized as in Fig. 3. will increase. The resulting production terms for - ( u v ) are listed below: dU
/~U)
/0V)
/~?U)
/0V)
The first three terms on the right-hand side, as detected by the Q2-scheme, are visualized in Fig. 8. The last two terms are not shown because they were found to be less influential and without any distinct features. Note that these two terms can be combined into -Fg(Ow/~z) due to mass continuity. The spanwise velocity fluctuations w, as well as their gradients, were found to be relatively small after inspection of ensemble averages in the xy- and xz-planes. The production terms in Eq. (5) appear as products of one time-averaged and one ensemble-averaged quantity. The latter averages to zero in the long-time limit in the last four terms, while the former can be treated as a constant for a given y. The first term in Eq. (5) gave the strongest contribution to (Plz) because of rather intense v fluctuations. As can be inferred from Fig. 6b, (?v/?x I was substantial around the centre of the events, thereby resulting in high magnitude of the third production term. But because (?v/~x) changes sign, this term averages to zero in the streamwise direction. The second term exhibited the same characteristics but was less influential due to the fact that @u/?y) was relatively weak in the Q2 events. This term was, per definition, strong in the PQ2 events. The characteristic features of the other production terms were fairly independent of the detection method, except Vf(~?u/Ox) which was of relatively small importance. Ensemble averages of the terms involving pressure (which are the same as those that occur in the time-averaged sense) are plotted in Fig. 9. Figs. 9a and 9b display the pressure-strain correlations. In the case of -(p~?u/~yl, relatively strong contributions in the near-wall layer are seen to be associated with the Q2 events at the centreline. This is partly because the pressure can be regarded as a global variable (since it can be derived from an integral of its Poisson equation) and therefore correlations prevail over larger length scales. In addition, (u) exhibited large correlation lengths, as can be inferred from Fig. 6a. In the time-averaged sense, -(p~?u/~?y) was the most important contribution in the near-wall region. In the central region, however, this term became less influental because (p) changes sign around x = 0 and the term will thus be reduced when averaged in the x-direction. The
76
K.H. Bech, H.I. Andersson / Fluid Dynamics Research 18 (1996) 6 5 - 7 9
a)
t
1.0 [ 0.0
.................... ..:.,
1 f:: ':"':'
'"' "" '::'::'
b)
:" " :":
-1.0
c)
-1.0 .... ::::::::::::::::::::::::: . . . . . . . 1.0 / ,. ,---,,,--:::,-.-.-,,.
:::'--?. . . . . :
d)
1
-1.o -4.0-3.0-2.0-1.o o.o 1.o 2.0 3.0 4.0 Fig. 8. Conditionally averaged pressure terms obtained from Q2-detection. (a) - ( p Ou/~y); (b) - ( p Ov/~x); (c) (u ~ p/~y); (d) (v 3p/~x);. Contour increment 0.05. Non-dimensionalized as in Fig. 3.
-(p~v/Ox) was the most important pressure term in the central region. It acted as a sink for -(uv) because both (p) and @v/~x) changed sign around x = 0. The velocity-pressure gradient cor-
term
relations in Figs. 9c and 9d appeared to be very similar to the corresponding pressure-strain correlations, except that the near-wall effects were weaker. This is in accordance with the fact that a contraction of the time-averaged pressure terms, cf. Eq. (4), reduces the complexity of the dynamics of -Fg. By comparing Figs. 8c and 9c, an interesting feature arises. There seems to be a strong negative correlation between the production t e r m uZ(Ov/Ox) and (u~p/Oy). These two terms seem to cancel each other out locally because the number of contours were similar. An analogous observation is valid for the terms (v2)dU/dy and (vOp/~x). This pairing of the major source and sink terms is by no means obvious.
4. Final discussion and concluding remarks The present result for the mean shear parameter, S = 0.22, compared favourably with the values 0.21 and 0.20 which Tillmark and Alfredsson (1993) deduced from the measurements of Aydin and Leutheusser (1991) and their own experimental results, respectively. These investigations were carried out at approximately the same Reynolds number. At the somewhat higher Re = 3000, the mean velocity profile obtained in the DNS of Lee and Kim (1991) showed that S < 0.20, although the authors claimed that their value of S was "about 30% of the value in the equivalent laminar flow with a linear velocity profile". It is interesting to recall that the pioneering experiments of Reichardt (1956) showed that the dimensionless slope in the central region of the Couette flow decreased with Re, and that Busse (1970), by means of a variational method, found the lower bound for infinite Re to be 0.25. The present finding and the experimental data collected by Tillmark and Alfredsson (1993) obviously do not support Busse's conclusion.
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79
77
If one assumes -V~ to be constant in a certain region, the viscous shear stress must also remain constant according to Eq. (1). This moreover implies that the production rate of turbulent kinetic energy, Pk, should be constant in that region. None of these assumptions were in accordance with the present results for the core region of the Couette flow. The viscous shear stress was not negligible and exhibited a relatively large variation, while the relative variation of the turbulent shear stress was modest. The variation of the mean velocity profile, as derived by von Karman (1937), showed reasonable agreement with the present results. Von Karman adopted the local equilibrium hypothesis, which turned out to be a reasonable first-order approximation because turbulent diffusion of kinetic energy was small. However, the central region of the Couette flow at low Reynolds numbers cannot be considered as truly homogeneous because the mean shear and some one-point statistics vary substantially. The Couette flow simulation (DNS) by Lee and Kim (1991) revealed a persistent vortical seeondary flow pattern in the form of counter-rotating streamwise vortices which had not been observed experimentally. The present simulation did not support the possible existence of streamwise vortices in the central region of turbulent Couette flow. The instantaneous structure of the flow field resembled large, elongated regions of altemating u fluctuations in combination with localized peaks in i v , and the ensemble averages obtained with Q2 detection exhibited simple spatial structures with length scales typical of the turbulence statistics. In this context, it may be interesting to recall that the DNS of homogeneous shear flow by Lee et al. (1990) indicated that a high shear rate alone is sufficient for generation of streaky structures, and that the presence of a solid boundary is not necessary. However, the present value 5.7 of the dimensionless shear rate parameter 2(dU/dy)k/e at the centreline of the Couette flow classifies the core region as a low-shear-rate flow in which streak-like structures should not be expected. There was no one-to-one correspondence between Q2 and PQ2 events and none of them could be considered as a subset of the others. In some of the PQ2 events, generation and destruction of the negative uv-correlation probably occurred simultaneously. Some of the Q2 events could be caused by other localized production terms, see Eq. (5), and by local positive correlations between velocity and pressure gradient. These differences may partly be due to the different detection methods and the adopted threshold values. The ensemble-averaged generation and annihilation terms were significantly correlated and with opposite signs, i.e. production and velocity-pressure gradient destruction of - u v occurred simultaneously. Insofar as second-moment turbulence modelling is concerned, the traditional way of handling correlations involving pressure is a decomposition into a pressure diffusion dlP2 and a pressure-strain correlation q~ij, i.e. in accordance with Eq. (3). The classical physical interpretation of the latter correlation, namely that the traceless tensor ~bij is responsible for an intercomponent energy transfer, involves only its diagonal components; cf. the pictorial arguments of Hinze (1975). The off-diagonal components have, on the other hand, traditionally been treated as the by-product of the decomposition and subsequent modelling of the pressure terms. By means of a coordinate transformation so that the coordinate axes became the principal axes of the Reynolds stresses, Hinze demonstrated that ~b~2 and -~-~ have opposite signs. Although Hinze's conclusion is in accordance with the present results (cf. Fig. 3), we do not accept his reasoning. By considering the shear stress, the traditional decomposition of the pressure terms does not necessarily ease the search for a physical interpretation. To this end the contraction in Eq. (4) should be adopted, and it has already been shown that vDp/3x represents the major contribution to the velocity-pressure gradient term H12 at the centreline.
78
K.H. Bech, H.I. Andersson/Fluid Dynamics Research 18 (1996) 65-79
Let us now consider the Q2 events depicted in Fig. 6. These events represent lumps of fluid moving in the positive y-direction (v > 0) with a streamwise velocity lower than their surroundings (u < 0). Thus, from the viewpoint of the faster moving surrounding fluid, the slowly propagating lumps are felt as obstacles. Since the excess velocity of the surrounding fluid is in the positive x-direction, a high-pressure zone should be expected on the upstream side of the lumps and a lowpressure region on the downstream side, i.e. analogous to the well-known pressure distribution around a solid sphere in a parallel stream. This conceptual model is supported by the observed pressure field in Fig. 6d. Moreover, the lump of fluid with velocity v > 0 should therefore be associated with a negative value of Op/~x so that vdp/~x < 0. This simple physical interpretation is thus in full accordance with the averaged velocity-pressure gradient term vOp/~x in Fig. 4. The Q2 events will, of course, also contribute favourably to -uv. The strong events considered here, moving in a background mean shear dU/dy, give rise to a significant positive correlation (v2)dU/dy. Thus, the Q2 events both create and destroy the correlation - u v . Both ensemble averages and time-averaged statistics showed that the two correlations, (vZ)dU/dy and (vOp/~x), were of crucial importance for the dynamics of - u v in the central region. Because the time-averaged results were partly reproduced by the ensemble averages, it can be concluded that the conditional sampling procedures were able to detect events representative for the structure of the flow. This implies that extremely strong events capture important features of the dynamics of - u v . In these events (v2) exhibited a peak value near that of the streamwise fluctuations (u2). In pressure-driven Poiseuille flow, however, such events seem less likely to occur because vz there exhibits a local minimum at the centreline. In the Poiseuille flow, near-wall structures, such as those captured by the VISA-technique, may be responsible for the major part of the turbulent shear-stress generation. Very similar structures were found in the near-wall region of the shear-driven Couette flow (see Bech et al., 1995). In the central region, however, different processes, some of which are described above, were responsible for generating and maintaining the high and almost uniform value of the turbulent shear stress in the Couette flow.
Acknowledgements This work has received support from The Norwegian Supercomputing Committee (TRU) through a grant of computing time. We are thankful to Professor P.-A. Krogstad at the Norwegian University of Science and Technology in Trondheim for commenting upon the results in Section 3.2. Dr. N. Tillmark at the Royal Institute of Technology in Stockholm made some valuable comments upon the two-point correlations. We are also grateful to the referees for their comments.
References Afzal, N. (1993) Asymptotic analysis of turbulent Couette flow, Fluid Dyn. Res. 12, 163. Andersson, H.I., K.H. Bech, and R. Kristoffersen(1992) On diffusionof turbulent energy in plane Couette flow, Proc. R. Soc. Lond. A 438, 477. Andersson, H.I. and B.A. Pettersson (1994) Modelling plane turbulent Couette flow, 25th AIAA Fluid Dynamics Conf., Paper 94-2342; Int. J. Heat Fluid Flow 15, 447. Aydin, E.M. and H.J. Leutheusser(1991) Plane-Couetteflow between smooth and rough walls, Exp. Fluids 11, 302.
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Bech, K.H. and H.I. Andersson (1994) Very-large-scale structures in DNS, in: P.R. Voke, L. Kleiser and J.-P. Chollet, eds., Direct and Large-Eddy Simulation I (Kluwer, Dordrecht) p. 13. Beth, K.H., R. Kristoffersen and H.I. Andersson (1993) Inner-layer velocity statistics in plane Couette flow, Int. Conf. Near-Wall Turbulent Flows, Tempe, Arizona, p. 317. Bech, K.H., N. Tillmark, P.H. Alfredsson and H.I. Andersson (1995) An investigation of turbulent plane Couette flow at low Reynolds numbers, J. Fluid Mech. 286, 291. Busse, F.H. (1970) Bounds for turbulent shear flows, J. Fluid Mech. 41,219. El Telbany, M.M.M. and A.J. Reynolds (1982) The structure of turbulent plane Couette flow, ASME J. Fluids Eng. 104, 367. Gavrilakis, S., H.M. Ysai, P.R. Voke and D.C. Leslie (1986) Large-eddy simulation of low Reynolds number channel flow by spectral and finite difference methods, Notes Numer. Fluid Mech. 15, 105. Gersten, K. (1985) The turbulent Couette flow from asymptotic theory view point, in: G.E.A. Meier and F. Oberrneier, eds., Lecture Notes in Physics, Flow of Real Fluids (Springer, Berlin) p. 219. Groth, J. (1991) Description of the pressure effects in the Reynolds stress transport equations, Phys. Fluids A 3, 2276. Hinze, J.O. (1975) Turbulence, 2nd ed. (McGraw-Hill, New York) p. 325. Karman, T. von (1937) The fundamentals of the statistical theory of turbulence, J. Aeronaut. Sci. 4, 131. Kim, J., P. Moin and R. Moser (1987) Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech. 177, 133. Kristoffersen, R., K.H. Bech and H.I. Andersson (1993) Numerical study of turbulent plane Couette flow at low Reynolds number, Appl. Sci. Res. 51, 337. Lee, M.J. and J. Kim (1991) The structure of turbulence in a simulated plane Couette flow, Proc. 8th Symp. on Turbulent Shear Flows, Munich. Lee, M.J., J. Kim, and P. Moin (1990) Structure of turbulence at high shear rate, J. Fluid Mech. 216, 561. Lu, S.S. and W.W. Willmarth (1973) Measurements of the structure of the Reynolds stress in a turbulent boundary layer, J. Fluid Mech. 60, 481. Lund, K.O. and W.B. Bush (1980) Asymptotic analysis of plane turbulent Couette-Poiseuille flows, J. Fluid Mech. 96, 81. Mansour, N.N., J. Kim and P. Moin (1988) Reynolds-stress and dissipation-rate budgets in a turbulent channel flow, J. Fluid Mech. 194, 15. Missimer, J.R. and L.C. Thomas (1983) Analysis of transitional and fully turbulent plane Couette flow, J. Lubr. Tech. ASME 105, 364. Papavassiliou, D.V. (1993) Direct numerical simulation of plane Couette flow, MS Thesis, Univ. of Illinois, UrbanaChampaign. Reichardt, H. (1956) Ober die Geschwindigkeitsverteilung in einer geradlinigen turbulenten Couettestr6mung, Z. Angew. Math. Mech. 36, 26. Reichardt, H. (1959) Gesetzmfissigkeiten der geradlinigen turbulenten Couettestr6mung, Max-Planck-Institut ffir Str6mungsforschung. Schneider, W., R. Eder and J. Sehmidt (1990) Turbulent Couette flow: asymptotic vs. experimental data, in: Nayfeh et al,, eds., Proc. 3rd Int. Congress on Fluid Mechanics, IV, 1593. Tillmark, N. and P.H. Alfredsson (1993) Turbulence in plane Couette flow, Appl. Sci. Res. 51, 237.