JOT J O U R N AL O F TUR BULEN C E http://jot.iop.org/

Turbulent supersonic channel flow Richard Lechner, J¨ orn Sesterhenn and Rainer Friedrich†

Received 4 August 2000; online 29 January 2001

Abstract. The effects of compressibility are studied in low Reynolds number

turbulent supersonic channel flow via a direct numerical simulation (DNS). A pressure–velocity–entropy formulation of the compressible Navier–Stokes equations which is cast in a characteristic, non-conservative form and allows one to specify exact wall boundary conditions, consistent with the field equations, is integrated using a fifth-order compact upwind scheme for the Euler part, a fourth-order Pad´e scheme for the viscous terms and a third-order low-storage Runge–Kutta time integration method. Coleman et al fully developed supersonic channel flow at M = 1.5 and Re = 3000 is used to test the method. The nature of fluctuating variables is investigated in detail for the wall layer and the core region based on scatter plots. Fluctuations conditioned on sweeps and ejections in the wall layer are especially instructive, showing that positive temperature, entropy and total temperature fluctuations are mainly due to sweep events in this specific situation of wall cooling. The effect of compressibility on the turbulence structure is in many respects similar to that found in homogeneous shear turbulence and in mixing layers. The normal components of the Reynolds stress anisotropy tensor are increased due to compressibility, while the shear stress component is slightly reduced. Characteristic of the Reynolds stress transport is a suppression of the production of the longitudinal and the shear stress component, a suppression of all velocity–pressure–gradient correlations and most of the dissipation rates. Comparison with incompressible channel flow data reveals that compressibility effects manifest themselves in the wall layer only.

† Author to whom correspondence should be addressed. c 2001 IOP Publishing Ltd

PII: S1468-5248(01)17904-X

1468-5248/01/000001+21$30.00

JoT 2 (2001) 001

Fachgebiet Str¨ omungsmechanik, Technische Universit¨at M¨ unchen, Boltzmannstr. 15, 85748 Garching, Germany E-mail: [email protected]

Turbulent supersonic channel flow

Contents Introduction

2

2

A (p, ui, s)-form of the Navier–Stokes equations and its numerical integration

3

3

Statistical relations

6

4

Results 4.1 Comparison with Coleman et al’s data 4.2 On the nature of fluctuating variables . 4.3 Structural compressibility effects . . . . 4.3.1 Reynolds stress anisotropy. . . . 4.3.2 Reynolds stress budgets. . . . . . 4.3.3 Instantaneous structures. . . . .

5

Conclusions

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7 7 9 16 17 17 21 22

1. Introduction Fully developed incompressible turbulent flow in a plane channel, one of the simplest wallbounded turbulent flows, has been directly simulated, for the first time, by Kim et al [1] at a low Reynolds number, Reτ = uτ h/ν = 180 (uτ and h are the friction velocity and channel half-width, respectively). Recently, Moser et al [2] reported direct simulations of the same flow at Reynolds numbers up to Reτ = 590. Since the late 1980s many researchers used low Reynolds number data to obtain valuable insights into the structure of wall-bounded incompressible turbulence or to test/improve statistical turbulence models as well as subgrid-scale models for large eddy simulation. In 1995 Coleman et al [3] presented the first DNS of turbulent supersonic channel flow and investigated compressibility effects. In order to achieve streamwise homogeneity in the simulation, the authors replaced the mean axial pressure gradient by an axially uniform body force, which was adjusted in such a way that the total mass flux through the channel remained constant. The walls were cooled and kept at a constant temperature Tw . The global Mach and Reynolds numbers, based on bulk velocity um , wall speed of sound cw , bulk density ρm , channel half-width h and wall viscosity µw , were M = 1.5 and 3 and Re = 3000 and 4880. The Prandtl number P r and the ratio of specific heats γ for air had the values 0.7 and 1.4, respectively. The dynamic viscosity varied as the 0.7th power of the temperature T . A density, velocity and temperature form of the Navier–Stokes equations was investigated using a Fourier–Legendre spectral discretization along with a hybrid implicit–explicit third-order four-substep timeadvance algorithm. The computational domain had the size (Lx , Ly , Lz ) = (4π, 4π/3, 2)h in streamwise, spanwise and wall-normal directions and was discretized by (144, 80, 119) collocation points, respectively. This corresponded to a grid spacing in the lower Mach number case (M = 1.5, Re = 3000) of ∆x+ ≈ 19 and ∆y + ≈ 12. The first collocation point in the wall-normal direction was at z + ≈ 0.1 and the first ten points were within z + ≈ 8. The DNS results of Coleman et al [3] revealed that the isothermal-wall boundary condition causes a flow that is strongly controlled by wall-normal gradients of mean density and temperature, to the point that most of the near-wall ρ and T fluctuations are the result of solenoidal ‘passive mixing’ by the turbulence. In other words, the dominant compressibility effect was found to be due to mean property variations so that the van Driest transformation of the mean axial velocity and a Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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Turbulent supersonic channel flow

2. A (p, ui, s)-form of the Navier–Stokes equations and its numerical integration In incompressible turbulence, vorticity fluctuations play an important role and vortex stretching is their key generation mechanism. Compressible turbulence is also characterized by vorticity fluctuations, but these can interact with entropy and pressure fluctuations. Kovasznay [7] denoted small-amplitude pressure (p), entropy (s) and vorticity (ωi ) fluctuations as ‘modes’ of compressible turbulence. It is worthwhile representing these ‘modes’ as directly as possible by corresponding transport equations. We favour a pressure, velocity and entropy formulation of the Navier–Stokes equations instead of a form involving ωi . Using the vorticity transport equations has the disadvantage that the two vorticity components tangential to a wall do not allow for exact wall boundary conditions. The continuity equation is converted into a pressure transport equation, replacing the material derivatives of density ρ by derivatives of pressure and entropy. For thermally perfect gases, the continuity equation takes the form of the desired pressure–transport equation: 1 Dp 1 Ds ∂uj =0 − + 2 ρc Dt cp Dt ∂xj

(1)

where c = (γp/ρ)1/2 is the speed of sound, γ is the ratio of specific heats at constant pressure, cp , and constant volume, cv . D(·)/Dt stands for the material derivative following the flow uj : ∂ ∂ D := + uj . Dt ∂t ∂xj Introducing variables of the form 
1 ∂p ∂u ± ± X := (u ± c) ρc ∂x ∂x

(2)

(3)

which describe the propagation of ‘waves’ in the positive and negative coordinate directions (here the x-direction), Sesterhenn [8] has cast the compressible Navier–Stokes equations in the following characteristic, non-conservative form:
 ∂p p ∂s ρc + − + − + − s s s +X +Y +Z (4a) = − ((X + X ) + (Y + Y ) + (Z + Z )) + ∂t 2 Cv ∂t

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scaling of the root mean square (rms) fluctuations by local values of density and viscosity proved successful. Further investigations into intrinsic compressibility effects of Coleman et al’s DNS data [3] by Huang et al [4] have demonstrated that explicit compressibility terms such as the pressure-dilatation correlation and the compressible dissipation rate, appearing in the turbulent kinetic energy balance equation, are negligibly small in supersonic channel flow and that neither the strong Reynolds analogy (SRA) suggested by Morkovin [5], nor the extended SRA by Cebeci and Smith [6] apply to non-adiabatic flow. Huang et al [4] finally provided a more general representation of the analogy which matches the DNS data very well. Despite the knowledge on compressibility effects in turbulent flows along cooled walls accumulated in [3, 4] from DNS data, many questions are still open. The present paper answers questions on the effect of compressibility on the pressure–strain correlation and the dissipation rate tensors appearing in the Reynolds stress budgets. Similarities to corresponding effects in mixing layers and homogeneous shear turbulence are worked out.

Turbulent supersonic channel flow

 1 ∂τ1j 1 + (X − X − ) + Y u + Z u + 2 ρ ∂xj 
1 ∂τ2j 1 + ∂v v − v = − X + (Y − Y ) + Z + ∂t 2 ρ ∂xj
 1 + 1 ∂τ3j ∂w w w − = − X + Y + (Z − Z ) + ∂t 2 ρ ∂xj
 ∂qi R ∂s s s s = −(X + Y + Z ) + − +Φ ∂t p ∂xi ∂u =− ∂t




with the abbreviations




:= (u + c)

Y

Y u := v

∂v ∂x

+

Y w := v


Z := (w + c)

∂w ∂y

1 ∂p ∂w + ρc ∂z ∂z

(4e)

X

X w := u

∂u ∂y

(4d)



∂w ∂x 
1 ∂p ∂v + := (v + c) ρc ∂y ∂y

X v := u +

1 ∂p ∂u + ρc ∂x ∂x

(4c)

−


:= (u − c)

X s := u Y

−




:= (v − c)

Y s := v 

∂s ∂x

−

1 ∂p ∂u − ρc ∂x ∂x

1 ∂p ∂v − ρc ∂y ∂y





∂s ∂y

Z := (w − c)




1 ∂p ∂w − ρc ∂z ∂z



∂u ∂v ∂s Z v := w Z s := w . (5) ∂z ∂z ∂z (u, v, w) = (u1 , u2 , u3 ) are the cartesian velocity components. The viscous stress tensor τij and the heat flux vector qi are defined by
 ∂uj 2 1 ∂ui (6) sij = + τij = 2µsij − µskk δij 3 2 ∂xj ∂xi Z u := w

∂T . ∂xi The dissipation rate Φ reads qi = −λ

(7)

Φ = τij sij .

(8)

Effects of bulk viscosity are neglected. The thermal equations of state p = ρRT

R = cp − cv

and the following power law for dynamic viscosity µ and heat conductivity λ namely 0.7
cp T µ = λ=µ µref Tref Pr

(9)

(10)

close the set of equations. The Prandtl number P r is assumed constant. One advantage of this special form of the Navier–Stokes equations is that boundary conditions can be derived consistently with the equations, without referring to one-dimensional inviscid approximations as proposed by Poinsot and Lele [9]. To give an example we discuss Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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X

+

(4b)

Turbulent supersonic channel flow

Z-

z

X X AAAAAAAAAAAAAAA AAAAAAAAAAAAAAA AAAAAAAAAAAAAAA Z+ +

-

x

Figure 1. Illustration of boundary treatment at a solid wall.

u=v=w=0

T = Tw = constant

at z = 0.

(11)

These imply ∂v ∂w ∂T ∂u = = = =0 at z = 0. (12) ∂t ∂t ∂t ∂t Equations (4a)–(4e) and (5) are used to derive unknown variables of type (3) at the wall. Figure 1 illustrates that Z + is a quantity that has to be specified at a wall. The other expressions are evaluated in the interior of the domain. From (4a) and (4d) we obtain ρc ∂p p ∂s = − (Z + + Z − ) + ∂t 2 cv ∂t ∂τ 2 3j Z+ = Z− + ρ ∂xj

(13) (14)

at z = 0. A second expression which relates pressure and entropy is derived from the Gibbs’ fundamental relation for an isothermal wall: ∂s R ∂p =− at z = 0. (15) ∂t p ∂t Combining (13) and (15) provides two conditions for p and s which show explicitly that p and s evolve in time at a wall if the flow field is time dependent, namely R ∂s = (Z + + Z − ) (16) ∂t 2c p ∂p = − (Z + + Z − ) (17) ∂t 2c at z = 0. Z + follows from (14). The special form of the Navier–Stokes equations, (4a)–(4e) and (5), suggests the use of an upwind discretization of wave-like terms. To this end the compact fifth-order upwind scheme ‘CULD’ (Compact Upwind with Low Dissipation) of Adams and Shariff [10] was used. Its resolution properties come close to those of spectral schemes. On the other hand, asymmetric coefficients generate the small amount of dissipation which is needed to suppress numerical instabilities that may be caused by unresolved high wavenumbers. An explicit filtering of data can thus be avoided. Spatial derivatives appearing in the viscous and heat conduction terms are discretized with a symmetric compact fourth-order scheme of Lele [11]. The solution is advanced in time with a third-order ‘low-storage’ Runge–Kutta scheme, as proposed by Williamson [12].

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the boundary conditions for a solid, isothermal wall and assume the x-coordinate to coincide with the stationary wall, while the z-coordinate is perpendicular to it. Then the kinematic and thermal boundary conditions are

Turbulent supersonic channel flow

3. Statistical relations In what follows, the overbar and the tilde are used to denote Reynolds- and Favre-averaged quantities, respectively. If appropriate, the velocity vector is decomposed in two ways: ui = u ¯i + ui = u ˜i + ui

u ˜i = ρui /¯ ρ.

(18)

 w  = 0 τ¯xz = u


at z = h.

(22)

Integrating the enthalpy balance in a similar way (see [4]), leads to a relation between the heat flux qw into the wall and the total pressure work done:   h ∂T  ∂ p¯ dz = −um τw . qw = −λw = u ˜ (23)  ∂z ∂x z=0

0

It immediately shows that an increase in the Mach number, while keeping the wall temperature constant, increases the necessary wall cooling rate and the near-wall temperature gradients. For fully developed, statistically steady channel flow, the Reynolds stress transport equations are: • ρu2 /2 − balance 0=




 d˜ u d  2   − ρu w /2 − u τxz

 dz dz



 w  −¯ ρu




Pxx

+u

T Dxx + V Dxx

• ρv 2 /2 − balance  dτyz d  2   +v  ρv w /2 − v  τyz 0=− dz

 dz T Dyy +V Dyy

T Dzz +P Dzz +V Dzz

P Sxx

Mxx

 ∂v   ∂v +p − τyj ∂y ∂x   j

Myy

• ρw2 /2 − balance  d  3  ρw /2 + w p − w τzz 0=− dz



  ∂u ∂ p¯ dτxz  ∂u + − +p − τxj ∂x dz ∂x ∂xj

 

P Syy


+w

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DSxx

(25)

DSyy

  ∂ p¯ ∂τzz ∂w  ∂w + − +p − τzj ∂z ∂z ∂z ∂xj

  Mzz

(24)

P Szz

(26)

DSzz

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Similarly, Reynolds and Favre fluctuations of the temperature are defined. The mean axial pressure gradient that drives the channel flow is replaced by a body force, namely ρ¯ τw ∂ p¯ = ρ¯f˜x = (19) − ∂x ρm h to achieve streamwise homogeneity of the flow in the simulation and to allow for comparison with Coleman et al’s data [3]. The alternative ∂ p¯ τw − = f¯x = (20) ∂x h has also been tested in a DNS, but will not be discussed here. It provides the expected linear profile of the total shear stress and leads to results which are quite close to those based on relation (19), where τw denotes the wall shear stress. The second part of (19) is obtained by integrating the axial momentum balance  ∂   w  + ρ τ¯xz − ρ¯u
¯f˜x (21) 0= ∂z from the wall (z = 0) to the symmetry plane (z = h) and using the symmetry conditions

Turbulent supersonic channel flow

• ρu w − balance

 d˜ u d   2

2  − w τ  − ρu w + u p − u τzz 0 = −¯ ρw xz

  dz dz Pxz

T Dxz +P Dxz +V Dxz

 
∂ p¯ ∂τzz ∂ p¯ ∂τxz + + w − + + u − ∂z ∂z ∂x ∂z





+p

∂u

∂w



Mxz

   ∂u − τ  ∂w . −τzj + xj ∂z ∂xj ∂x ∂xj



 −DSxz

 w  , contains a negative production term P , The balance of the Reynolds shear stress, u
xz  w  is of a negative sign, which leads to a positive when the shear rate d˜ u/dz is positive. Then u
production rate Pxx of streamwise fluctuations. The spanwise and wall normal fluctuations, v  and w obtain their kinetic energy from the pressure–strain correlations P Syy and P Szz . The 2 component obtains its energy primarily via production, while the other two fact that the u components obtain their energy via the secondary process of redistribution, explains why the Reynolds stresses are anisotropic. The remaining terms in these equations describe the mass flux variation, Mij ; turbulent, viscous and pressure transport, T Dij , V Dij and P Dij ; and, finally, dissipation, DSij . The importance of these terms will become clear from figures 17–21, below.

4. Results 4.1. Comparison with Coleman et al’s data Free parameters of the simulated channel flow are the Mach number M , the Reynolds number Re, the Prandtl number P r and the ratio of specific heats γ. We recall their definitions M = um /cw P r = µw cp /λw

Re = ρm um h/µw γ = cp /cv µw ∼ Tw0.7

(28) (29)

and specify their values in table 1. The computational domain is the same as that of [3], see the introduction. It is resolved by (144 × 80 × 129) points in the (x, y, z) directions. While the grid is equidistant in the (x, y) directions, the points are distributed in the wall-normal direction according to the transformation:   tanh[β(1 − 2ξ)] . (30) z =h 1− tanh(β) Here ξ ∈ [0; 1] represents the uniform grid distribution and z is its non-uniform counterpart. A value of β = 1.55 leads to a first grid point at z1+ = z1 uτ /νw ≈ 0.985 and a tenth point at + z10 ≈ 12.073. The friction velocity is defined by uτ = (τw /ρw )1/2 . The maximum grid spacing in the core region is ∆z + = 5.84. The DNS was started from an instantaneous turbulence field that Gary Coleman kindly provided. It was interpolated on the present grid. The time step was controlled based on a linear stability criterion. The computation was performed on one node of a Fujitsu VPP 700 with a code performance of 500 MFlops. The statistical data presented below are based on 300 instantaneous three-dimensional fields extracted from time series which cover 36 problem times h/uτ after an initial transient of 15h/uτ had elapsed. The averaging time of 36h/uτ allows Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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P Sxz

(27)

Turbulent supersonic channel flow

Table 1. Flow parameters. M

Re

Pr

γ

1.5

3000

0.7

1.4

30

25

+

15

10

5

0 1

10

100

1000

Figure 2. Van Driest transformed velocity: et al [3] and - - - - - - log law.

z+

present case, – – – Coleman

2 1.8 1.6

T /Tw

✟ ✟

1.4 1.2 1

❍ ❍

ρ/ρm

0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

Figure 3. Mean density and temperature profiles: Coleman et al [3].

z/h

present case and – – –

slow near-wall fluid to pass the channel (of length 4πh) 2.86 times. One problem time, which corresponds to somewhat more than 8000 time steps, consumed 38.7 CPU hours. Figures 2–7 compare the van Driest transformed mean velocity, the mean density and temperature, the Reynolds shear stress, the rms-density and temperature fluctuations and 1/2  the total rms-vorticity fluctuations, (ωtotal )rms = (ωi ωi ) , with the results of [3]. The generally good agreement between both computations underlines the suitability of the present numerical method for the DNS of compressible turbulence. The small differences in the vorticity Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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uvD

20

Turbulent supersonic channel flow 0.004

2

ρu''w''/ρmum

0.002

0

-0.002

-0.004 0.5

1

1.5

Figure 4. Reynolds shear stress: [3].

2

z/ h

present case and – – – Coleman et al

0.06

0.05

ρrms/ρ

0.04

0.03

0.02

0.01

0 0

0.5

1

Figure 5. Rms density fluctuations: al [3].

1.5

2

z/ h

present case and – – – Coleman et

fluctuations of the buffer layer can certainly be removed by increased spanwise resolution. This is concluded from the high wavenumber behaviour of one-dimensional spanwise energy spectra. 4.2. On the nature of fluctuating variables Supersonic channel flow is achieved only if most of the heat generated by dissipation is removed through the walls, i.e. by wall cooling. Hence the walls are colder than the bulk of the flow. Since the mean flow is approximately isobaric (mean pressure variations normal to the wall reflect the behaviour of the wall-normal turbulence intensity), the mean density has its maximum at the wall and the normalized temperature and density gradients are practically equal in magnitude, but opposite in sign: 1 ∂ T¯ 1 ∂ ρ¯ − ¯ ≈ . (31) ρ¯ ∂z T ∂z

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0

Turbulent supersonic channel flow 0.06

Trms/T

0.05

0.04

0.03

0.02

0.01

0

0.5

1

1.5

Figure 6. Rms temperature fluctuations: et al [3].

z/ h

2

present case and – – – Coleman

0.6

(ω'total)rms

0.5

0.4

0.3

0.2

0.1

0 0

50

100

150

200

z+

 Figure 7. Total rms vorticity fluctuations, (ωtotal )+ rms : – – – Coleman et al [3].

present case and

The observed near-wall variations (figure 3) are rapid and qualitatively different to those found in adiabatic-wall boundary layers [13, 14]. It can be shown that the steep mean density gradients are the cause for the surprisingly high level of density fluctuations in the buffer layer (see figure 5). The transport equation for ρ2 takes the following form in fully developed channel flow: 0 = −2ρ w

∂uj ∂uj d¯ ρ d 2  − 2¯ ρ ρ − ρ 2 − ρ w . dz ∂xj ∂xj dz

(32)

The first term, which is dominant, describes production by mean density gradients, the second and third terms produce density fluctuations via dilatation fluctuations and the last term reflects turbulent transport. In this flow there is no production by mean dilatation. The turbulent mass flux ρ w = −¯ ρw peaks in the buffer layer [4]. The negative mean density gradient and the positive turbulent mass flux in the buffer layer lead to the main positive source Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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term in (32), namely the production of density fluctuations. The contributions of terms involving dilatation fluctuations are negligible near the wall. In the symmetry plane, on the other hand, the first production term vanishes. The remaining level of density fluctuations must therefore be due to dilatation fluctuations. The transport term vanishes in this plane. After a few preliminary remarks, we analyse the fluctuating variables based on scatter plots, and point out the differences between the near-wall and the core flow. Gaviglio [15] and Rubesin [16] noted the failure of Morkovin’s [5] SRA in non-adiabatic flows. The SRA neglects total temperature fluctuations and leads to the following relation:  T ¯ 2u ≈ −(γ − 1) M u ¯ T¯

(33)

with α = 1 and 1.34 according to [15] and [16], respectively. T¯0 is the mean total temperature. Huang et al [4] have shown that c approximately equals the turbulent Prandtl number. Following Rubesin [16], they assumed that the thermodynamic fluctuations behave in a polytropic manner n ρT  p ρ =n = p¯ ρ¯ n − 1 ρ¯T¯

(35)

and came up with a model for the mean Favre fluctuations, which for n = 0 is in excellent agreement with the near-wall behaviour of supersonic turbulent channel flow as predicted by DNS [3]. Figure 8 shows scatter plots of temperature against density and velocity fluctuations. In these plots fluctuations at each second point of the (144 × 80) computational points of a plane parallel to the channel walls are displayed. In planes close to the wall (plots on the left) the fluctuations are conditioned on sweeps (u > 0, w < 0, red) and ejections (u < 0, w > 0, green). In the symmetry plane such a distinction does not make sense. It is interesting to note that sweeps/ejections carry mostly positive/negative temperature fluctuations, respectively. A closeup for small amplitudes in figure 9, however, shows that small negative/positive temperature fluctuations do occur in sweeps/ejections as well. The scatter plot for T  against u at z + = 9 contradicts Morkovin’s relation (33) as expected (positive slope) and supports equation (34) in the sense that the suggested value of (T  /T¯)/(u /¯ u) = 0.12 together with the gradient ∂ T¯0 /∂ T¯ = 2.08 at z + = 9 favour a value of α = 1.05. It further indicates a turbulent heat flux in the mean flow direction which, of course, is energetically inactive. The turbulent heat flux derived from (T  , w ) fluctuations, on the other hand, transports energy towards the wall (the wall is a heat sink). Linearizing the second equality of (35) a relation of the form (n − 1)

T ρ ≈ ¯ ρ¯ T

(36)

results, which for n = 0 implies that pressure fluctuations are unimportant and that the correlation coefficient R(ρ, T ) is minus one: R(ρ, T ) =

ρrms T¯ ρ T  = −1. =− ρrms Trms Trms ρ¯

(37)

The scatter plot in figure 8 supports this result for the near-wall region. Figure 10 confirms that pressure fluctuations are unimportant compared to density fluctuations and that the correlation coefficient R(ρ, p) is close to zero. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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¯ =u where M ¯/¯ c is the local Mach number. These authors have independently derived a relation of the form T  /T¯ 1 ≈ (34) 2  ¯ u /¯ (γ − 1)M u α(∂ T¯0 /∂ T¯ − 1)

Turbulent supersonic channel flow 0.02 0.1 0.01

T'/T

T'/T

0.05 0 -0.05

0 -0.01

-0.1 -0.02 -0.1

0

0.1

-0.05

0

0.05

ρ'/ρ

ρ'/ρ 0.02 0.1

T'/T

T'/T

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0.01

0.05 0 -0.05

0 -0.01

-0.1 -0.02 -1

0

1

-0.2

-0.1

u'/u

0

0.1

0.2

0.1

0.2

u'/u 0.02

0.1 0.01

T'/T

T'/T

0.05 0 -0.05

0 -0.01

-0.1 -0.02 -0.1

0

0.1

-0.2

-0.1

0

w'/u

w'/u

0.02

0.04

0.01

0.02

T0'/T0

T'/T

Figure 8. Scatter plots of temperature against density (top) and velocity fluctuations in planes parallel to the channel wall at z + = 9 (left) and in the symmetry plane (right).

0

0

-0.01

-0.02

-0.02

-0.04 -0.02

0

ρ'/ρ

0.02

-0.02

0

0.02

T'/T

Figure 9. Close-up of temperature against density and total temperature fluctuations in a plane close to the wall at z + = 9. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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0.15

0.06

0.1

0.04

0.05

0.02

ρ'/ρ

ρ'/ρ

Turbulent supersonic channel flow

0

0

-0.05

-0.02

-0.1

-0.04

-0.15

-0.06 -0.1

0

0.1

-0.1

p'/p

0

0.1

p'/p

Let us now assume linear relations between thermodynamic fluctuations of the form T ρ p  p ρ T  s = −(γ − 1) + ¯ = −γ + = γ ¯ − (γ − 1) cv ρ¯ ρ¯ p¯ p¯ T T obtained from the entropy definition s = cv ln(p/ργ )

(38)

(39)

and the linearized gas law ρ T  p = + ¯. (40) p¯ ρ¯ T Neglecting pressure fluctuations in the near-wall region with respect to density fluctuations in (38), leads to s ρ T = −γ = γ ¯ . cv ρ¯ T

(41)

The scatter plots of (s , ρ ), (s , T  ) and (s , p ), in figure 11, indeed confirm this law surprisingly well. An explanation of why the pressure fluctuations have such a low level can be given after inspection of the pressure variance transport equation [17] which, for fully developed channel flow, takes the form 0=

−2p u

∂uj ∂uj ∂ p¯ d  2  2 − 2γ p¯p − (2γ − 1)p − w p ∂x ∂xj ∂xj dz
 ∂ ∂T  λ + viscous terms. +2(γ − 1)p ∂xj ∂xj

(42)

It is the axial mean pressure gradient and its interaction with the pressure–velocity correlation that is responsible for the low level of pressure fluctuations (first term on the right-hand side) in the whole channel. An increase in global Mach number would certainly increase the amplitudes of the pressure fluctuations via an increased pressure gradient. Production due to mean dilatation, ∂ u ˜j /∂xj , which is important in compression zones such as shocks, does not exist in this flow and has dropped out of the equation. Terms two and three, accounting for the production by dilatation fluctuations, are unimportant. Term four is a transport term which does not contribute to the level of p2 . The last terms involve viscous transport and ‘dissipation’ effects. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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Figure 10. Density against pressure fluctuations in planes parallel to the channel walls at z + = 9 (left) and at the centreline (right).

Turbulent supersonic channel flow 0.15 0.02

0.1

0.01

s'/cv

s'/cv

0.05 0 -0.05

0 -0.01

-0.1

-0.02

-0.15 -0.1

0

0.1

-0.05

0

0.05

ρ'/ρ

ρ'/ρ 0.2

0.01

s'/cv

s'/cv

0.1

JoT 2 (2001) 001

0.02

0

0 -0.01

-0.1

-0.02 -0.2 -0.1

0

0.1

-0.02

-0.01

0

0.01

0.02

T'/T

T'/T 0.2 0.02 0.01

s'/cv

s'/cv

0.1 0

0 -0.01

-0.1

-0.02 -0.2 -0.1

0

0.1

-0.1

p'/p

0

p'/p

0.1

Figure 11. Entropy fluctuations against density, temperature and pressure fluctuations in planes parallel to the channel walls at z + = 9 (left) and at the centreline (right). Finally, we demonstrate that the total temperature fluctuations are non-negligible in the wall region of the present flow, in contrast to compressible flow along adiabatic walls. From the defining relation uu + (ui ui − ui ui )/2)/cp T0 ≈ T  + (¯ we obtain T¯ T0 = ¯ ¯ T0 T0



T ¯2 + (γ − 1)M T¯



− u2 u u2 + i 2 i u ¯ 2¯ u

(43)  .

(44)

¯u . In order to establish In these expressions the term ww is neglected compared to u the relative importance of the remaining terms in the channel we specify the necessary mean quantities for the two positions z + = 9 and 220 (centreline) in table 2. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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Turbulent supersonic channel flow

Table 2. Mean flow quantities for the two channel positions. z+

T¯/T¯0

¯ =u M ¯/¯ c

¯2 (γ − 1)M

9 220

0.930 0.687

0.582 1.507

0.136 0.908

0.12

0.1

T0,rms/T0

0.06

0.04

0.02

0 0

0.5

1

1.5

2

z/h

Figure 12. Rms total temperature fluctuations across the supersonic channel. From figure 12, which contains the profile of the rms total temperature fluctuations, we read peak values for T0,rms /T¯0 of 10% in the wall layer. The minimum value in the core is still 3%. Figure 13 presents scatter plots of T0 /T¯0 against temperature, velocity and kinetic energy fluctuations (ui 2 − u2 u2 ) (cf also figure 9). From these we conclude, taking the data of i )/(2¯ table 2 into account, that in the wall layer the contributions from temperature, velocity and kinetic energy fluctuations to the total temperature fluctuations are of equal importance. Only in the core region does a linear relation of the form ¯  T0 ¯2 T u (45) = (γ − 1) M ¯ T¯0 T¯0 u represent a permissible approximation. Until now, the discussion of fluctuations has focused on the near-wall region where most of the turbulent kinetic energy is produced by high shear rates and where density, temperature and total temperature fluctuations peak because of the strong mean density and temperature variations. In the core region we note a reduction in the level of density, temperature, entropy and velocity fluctuations by nearly an order of magnitude. Pressure fluctuations are low everywhere. The total temperature fluctuations decrease by a factor of three only from the buffer layer to the core region. It turns out that the pressure fluctuations are no longer small in the core, compared to the remaining thermodynamic fluctuations, and that they behave in a nearly isentropic manner, with n ≈ 1.49 (see equation (35)). Concerning the total temperature fluctuations, it is found that they are purely controlled by velocity fluctuations. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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0.08

Turbulent supersonic channel flow 0.4

0.1

0.3 0.05

T'0/T0

T'0/T0

0.2 0.1

0

0 -0.05 -0.1 -0.2 -0.1

0

-0.1 -0.02

0.1

-0.01

0

0.01

T'/T

T'/T 0.4

0.1

JoT 2 (2001) 001

0.3

T'0/T0

T'0/T0

0.05 0.2 0.1

0

0 -0.05 -0.1 -0.2 -1

0

-0.1 -0.2

1

u'/u

-0.1

0

0.1

0.2

u'/u

0.4

0.1

0.3

T'0/T0

T'0/T0

0.05 0.2 0.1

0

0 -0.05 -0.1 -0.2 -0.1

-0.1 0

0.1

0.2

0.3

0.4

(u'i2 - u'i2)/(2u2)

0

0.01

0.02

(u'i2 -

u'i2)/(2u2)

Figure 13. Total temperature fluctuations against temperature, velocity and kinetic energy fluctuations in planes parallel to the channel walls at z + = 9 (left) and at the centreline (right). 4.3. Structural compressibility effects It is interesting to contrast the present DNS data at M = 1.5 with the classical data of Kim et al [1] for an incompressible channel flow. Such a comparison has partly been done in [3, 4]. We go to a step further and discuss compressibility effects on the complete Reynolds stress transport. We start by pointing out the differences in the turbulence intensities and the Reynolds shear stress when the flow becomes compressible. Figure 14 shows profiles of (ρui uj /τw )1/2 for the three normal components and the profile of the Reynolds shear stress −ρu1 u3 /τw . This type of presentation is different from that used in [3, 4]. Obviously, the effect of compressibility is to increase the longitudinal component and to decrease the remaining three components. In the core region we note a collapse with the data for incompressible flow. Compressibility effects are Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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Turbulent supersonic channel flow 3

2.5

2

(ρu1 u1 /τw )1/2

1.5

(ρu2 u2 /τw )1/2 ❅ ❅

1

0.5

(ρu3 u3 /τw )1/2

0 0

0.5

1

z/h

Figure 14. Rms values of the normal components of the Reynolds stress tensor normalized with (τw )1/2 and of the Reynolds shear stress −ρu1 u3 /τw : – – – M = 0 [1] and M = 1.5. hence only observed in the wall layer, where the non-dimensional shear rate Sk/3 (cf figure 16) is high. 4.3.1. Reynolds stress anisotropy. The Reynolds stress anisotropy defined by aij = 2

ρui uj ρuk uk

2 − δij 3

(46)

is an important indicator of compressibility effects and appears in advanced turbulence models. For homogeneous shear turbulence, Sarkar [18] has conclusively shown that the reduction in the growth rate of the turbulence kinetic energy as the gradient Mach number Mg increases, is due to the reduction in the Reynolds shear stress anisotropy. This was later confirmed by Simone et al [19] for the early-time evolution of the flow based on rapid distortion theory. The decrease in a13 is accompanied by an increase in the diagonal components a11 , a33 [18]. These structural compressibility effects were recently studied in detail by Hamba [20] based on the complete set of Reynolds stress transport equations. Increased values of normal stress anisotropy with the convective Mach number were also observed in DNS studies of compressible mixing layers by Vreman et al [21] and Freund et al [22]. Figure 15 compares the Reynolds stress anisotropy for incompressible flow [1] with that for supersonic flow. Figure 16 shows profiles of the nondimensional shear rate Sk/3 (S = d˜ u/dz) and the gradient Mach number Mg = Sk 3/2 /(3 c¯). Both quantities have a very similar shape. The behaviour of the Reynolds stress anisotropies is in agreement with the observations in homogeneous shear turbulence and in mixing layers in the sense that an increase in the Mach number amplifies the normal stress anisotropies and reduces the shear stress anisotropy (or leaves it nearly unaltered). It is also found that the compressibility effect is weak in the log layer, where the flow is close to equilibrium (Sk/3 ≈ 3) and is pronounced in the buffer layer where the non-dimensional shear rate and the local gradient Mach number have their highest values (Sk/3 ≈ 24, Mg ≈ 3). 4.3.2. Reynolds stress budgets. In their analysis of compressibility effects in a supersonic channel flow, Huang et al [4] concentrated on energetic rather than structural compressibility Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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−ρu1 u3 /τw

❆ ❆

✁ ✁

Turbulent supersonic channel flow 1.2 1 0.8 0.6

——— –––– ----—· —· ····· –· –· –·

a 11 a 22 a 33 a 12 a 13 a 23

——— –––– ----·····

a 11 a 22 a 33 a 13

aij

0.4 0.2 0 -0.2 -0.4

-0.8 0

0.5

1

z/h

25

5

20

4

15

3

10

2

5

1

Mg

Sk/ε

Figure 15. Reynolds stress anisotropies for incompressible (M = 0 [1], red curves) and compressible flows (M = 1.5, green curves).

0

0 0

0.5

1

z/h

Figure 16. Local mean shear rate Sk/3 ( ) and gradient Mach number Mg ( – – – ) in supersonic channel flow (M = 1.5).

effects. They show that the compressible dissipation rate and the pressure dilatation correlation are negligibly small (and hence should be removed from the k equation when modelling supersonic wall-bounded turbulence). They have also demonstrated the reduction in k production as a consequence of compressibility. From these results it is yet unclear which mechanism dampens the production of turbulent kinetic energy. A more complete picture of compressibility effects can only emerge from an inspection of the budgets of all Reynolds stress components. It also means that a set of two-equation turbulence models can hardly provide satisfactory predictions of compressibility effects. Figures 17–20 show the terms in the budgets of the four Reynolds stresses presented in equations (24)–(27) and provide a comparison between incompressible [1] and compressible Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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JoT 2 (2001) 001

-0.6

Turbulent supersonic channel flow

Gai n

3

2

1

——— –––– ----—· —· ····· –· –· –·

P 11 TD 11 VD 11 M 11 VPG 11 DS11

——— –––– ----····· –· –· –·

P 11 TD11 VD11 VPG 11 DS11

Loss

0

-1

-2 0.1

0.2

0.3

0.4

0.5

0.6

z/ h

Figure 17. Terms in the budget of ρu2 1 /2 for incompressible (M = 0, red curves) and compressible flows (M = 1.5, green curves). 0.6

Gai n

0.4

0.2

——— –––– ----—· —· ····· –· –· –·

P22 TD 22 VD 22 M 22 VPG 22 DS22

——— –––– ----····· –· –· –·

P22 TD 22 VD22 VPG 22 DS22

Loss

0

-0.2

-0.4 0

0.1

0.2

0.3

0.4

0.5

0.6

z/ h

Figure 18. Terms in the budget of ρu2 2 /2 for incompressible (M = 0, red curves) and compressible flows (M = 1.5, green curves). channel flow. All terms have been non-dimensionalized with τw um /h = −qw /h, which is the best choice for this flow, cf [4]. Only the sums of pressure–strain and pressure–diffusion terms, i.e. the velocity–pressure–gradient correlations are shown. Mass-flux variations Mij do not exist in incompressible flow. The sum of all terms has not been shown in these figures, for clarity. However, it never exceeds values of 0.01, in agreement with the accuracy of the incompressible data. From these figures it is again clear that compressibility manifests itself mainly in the wall layer and leads to reductions in all production and velocity–pressure–gradient terms and most of the dissipation terms. In the core region of the channel, the effects of the compressibility are so small that the various profiles for M = 0 and M = 1.5 practically collapse. For convenience, the k budget is presented in figure 21, which as a main effect reveals the reduction in the production and dissipation rate. Both effects had been noted by Sarkar [18] for homogeneous shear turbulence. There is, however no explanation, until now, why the dissipation rate tensor is reduced in its amplitudes. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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JoT 2 (2001) 001

0

Turbulent supersonic channel flow 0.25

Gai n

0.2 0.15 0.1

——— –––– ----—· —· ····· –· –· –·

P 33 TD 33 VD 33 M 33 VPG 33 DS 33

——— –––– ----····· –· –· –·

P 33 TD 33 VD33 VPG 33 DS33

0.05 0

Loss

-0.05 -0.1 -0.15

0

0.1

0.2

0.3

0.4

0.5

0.6

z/ h

Figure 19. Terms in the budget of ρu2 3 /2 for incompressible (M = 0, red curves) and compressible flows (M = 1.5, green curves).

Gai n

1

0.5

——— –––– ----—· —· ····· –· –· –·

P13 TD 13 VD 13 M 13 VPG 13 DS 13

——— –––– ----····· –· –· –·

P 13 TD 13 VD 13 VPG 13 DS13

Loss

0

-0.5

-1 0

0.1

0.2

0.3

0.4

0.5

0.6

z/ h

Figure 20. Terms in the shear stress budget, ρu1 u3 , for incompressible (M = 0, red curves) and compressible flows (M = 1.5, green curves). Figure 22 summarizes the effect of compressibility on the velocity–pressure–gradient (V P G) terms. It must be kept in mind that the 11 and 22 components equal the corresponding pressure–strain (P S) terms and that only V P G33 differs from P S33 by the pressure–diffusion (P D) term, P D33 . Since the pressure–dilatation correlation is small, figure 22 also gives insight into the redistribution of turbulent kinetic energy. Now, V P G11 is negative and the terms V P G22 and V P G33 are positive, hence the spanwise and wall normal components receive energy from the streamwise component. All the terms are suppressed in the wall region as a consequence 2 of compressibility. This means that the energy input into ρu2 2 and ρu3 is reduced. According 2 dampens the production term and thus the level of ρu u . to (27), a reduction in ρu2 3 = ρw 1 3 A suppressed shear stress in turn suppresses the production of ρu1 2 . Why do we then observe 1/2

an increase in (ρu1 2 /τw ) in figure 14 due to compressibility? The answer to this question is not obvious. It has to incorporate the reduction in the dissipation rate and in the redistribution term and, finally, the modifications in the viscous and turbulent diffusion terms. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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-0.2

Turbulent supersonic channel flow

Gain

3

2

1

——— –––– ----—· —· ····· –· –· –·

Pk TD k VD k Mk VPG k DSk

——— –––– ----····· –· –· –·

Pk TD k VD k VPG k DSk

Loss

0

-1

-2 0.1

0.2

0.3

0.4

0.5

0.6

z/ h

Figure 21. Terms in the kinetic energy budget for incompressible (M = 0, red curves) and compressible flows (M = 1.5, green curves). 0.8

Gai n

0.6

——— –––– ----·····

VPG 11 VPG 22 VPG33 VPG 13

——— –––– ----·····

VPG 11 VPG 22 VPG33 VPG13

0.4 0.2

Loss

0 -0.2 -0.4 -0.6 0

0.1

0.2

0.3

0.4

0.5

0.6

z/ h

Figure 22. Velocity–pressure–gradient tensor, V P Gij . Red curves denote incompressible flow (M = 0) and green curves compressible flow (M = 1.5). 4.3.3. Instantaneous structures. Based on contours of wall-normal vorticity, Coleman et al [3] illustrated that compressibility makes the near-wall streaks more coherent, which in turn increases the streamwise correlation length. The spanwise streak spacing, on the other hand, is roughly the same as in an incompressible flow. Our aim here is to point out another similarity between the channel flows at M = 0 and at M = 1.5. Figure 23 shows contour lines of axial velocity fluctuations, u1 , in the buffer layer (z + = 9). Red lines indicate sweeps and green lines indicate ejections. They appear alternatingly in the spanwise direction and transport high-speed fluid to the wall and low-speed fluid away from the wall. The perspective view of the isosurfaces of u1 in figure 24 provides an even more detailed insight into the shape of these structures. Besides sweeps and ejections this plot contains isosurfaces of the second invariant q of the velocity gradient tensor: q=−

1 ∂ui ∂uj 1 = (ωi ωi − 2sij sij ) 2 ∂xj ∂xi 4

Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

(47) 21

JoT 2 (2001) 001

0

Turbulent supersonic channel flow

4

y

3 2 1 0 0

5

10

x

which allows us to isolate vortices (q > 0). Figure 24 clearly shows that these vortices are generated in the shear layers between sweeps and ejections, the latter representing forward–downward, respectively backward–upward, motion in a frame of reference moving with the mean flow. Contour lines of sweeps/ejections and vortices (white) in a plane perpendicular to the main flow further underline the position of the vortices as being between sweeps and ejections. Such plots have, for the first time, been presented by Dubief in his thesis [23] for incompressible channel flow. They underline the similarities between instantaneous structures of both flows.

5. Conclusions A new formulation of the compressible Navier–Stokes equations has been successfully applied to investigate turbulent supersonic channel flow via direct numerical simulation. It is based on transport equations for pressure, velocity and entropy and thus allows one to compute Kovasznay’s ‘modes’ of compressible turbulence more directly. Variables have been introduced in these equations which describe the propagation of ‘waves’ in the positive and negative coordinate directions and allow one to derive exact wall boundary conditions which are consistent with the equations and avoid one-dimensional inviscid approximations. In order to test the new code, we have selected Coleman et al’s [3] supersonic channel flow at M = 1.5 and Re = 3000. Comparison with their spectral DNS data shows good agreement and underlines the efficiency of the new method. An investigation of the nature of velocity, pressure, entropy, density, temperature and total temperature fluctuations based on scatter plots and transport equations has provided the following insights. • Density fluctuations are produced by the mean density gradient in the wall layer and hence peak there. • Pressure fluctuations are produced by the mean axial pressure gradient and have a low level everywhere in the channel. While they can be neglected in the wall layer, they are of the same order of magnitude (when normalized by p¯) as the normalized density, temperature and entropy fluctuations and follow a nearly isentropic law. • Temperature fluctuations are perfectly correlated with density fluctuations in the wall layer. Conditioned on sweeps and ejections they show that sweeps/ejections preferably carry positive/negative temperature fluctuations, respectively, in this specific situation of wall cooling. Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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Figure 23. Contour lines of u /um at z + ≈ 9 (green: u /um < 0, red: u /um > 0).

Turbulent supersonic channel flow

JoT 2 (2001) 001 Figure 24. Top, contour lines of u /um and q at x = 11 (green, u /um < 0; red, u /um > 0; white, q > 0). Bottom, isosurfaces of u /um and q. (green, u /um = −0.15; red, u /um = 0.15; white, qh2 /u2m = 1.0). • Entropy fluctuations are strongly correlated with density and temperature fluctuations near the wall. The fact that entropy and pressure fluctuations are decoupled there underlines the lack of intrinsic compressibility effects and the idea of pure solenoidal mixing [3]. • Total temperature fluctuations reach maximum values of 10% of the mean total temperature in the wall layer and are due to temperature, velocity and kinetic energy fluctuations. In the core region, on the other hand, they are reduced by a factor of three, but there they are only linearly related to the velocity fluctuations. The effect of compressibility on the turbulence structure has been studied based on the Reynolds stress transport equations and on a comparison with the Kim et al data [1]. A normalization of the Reynolds stress tensor ρui uj by the wall shear stress, τw = ρw u2τ reveals that effects of compressibility appear in the wall layer only, at least at a moderate Mach number of 1.5. It is found that only the axial stress component is increased compared to its incompressible counterpart, whereas the other two normal and the shear stress components are suppressed. The Reynolds stress anisotropy tensor behaves in a way which is similar to the behaviour of compressible homogeneous shear turbulence [18] and of compressible mixing layers [21, 22]. An Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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Turbulent supersonic channel flow

Acknowledgments We are grateful to Gary Coleman for providing his instantaneous and statistical data and to the referees for their valuable comments.

References [1] Kim J, Moin P and Moser R 1987 Turbulence statistics in fully developed channel flow at low Reynolds number J. Fluid Mech. 177 133–66 [2] Moser R D, Kim J and Mansour N N 1999 Direct numerical simulation of turbulent channel flow up to Reτ = 590 Phys. Fluids 11 943–5 [3] Coleman G N, Kim J and Moser R D 1995 A numerical study of turbulent supersonic isothermal-wall channel flow J. Fluid Mech. 305 159–83 [4] Huang P G, Coleman G N and Bradshaw P 1995 Compressible turbulent channel flows: DNS results and modelling J. Fluid Mech. 305 185–218 [5] Morkovin M V 1962 Effects of compressibility on turbulent flows M´ecanique de la Turbulence ed A Favre (London: Gordon and Breach) pp 367–80 [6] Cebeci T and Smith A M O 1974 Analysis of Turbulent Boundary Layers (New York: Academic) [7] Kovasznay L S G 1953 Turbulence in supersonic flow J. Aeronaut. Sci. 20 657–82 [8] Sesterhenn J 2001 A characteristic-type formulation of the Navier–Stokes equations for high order upwind schemes Computers and Fluids 30 37–67 [9] Poinsot T J and Lele S K 1992 Boundary conditions for direct simulations of compressible viscous flows J. Comput. Physics. 101 104–29 [10] Adams N A and Shariff K 1996 A high-resolution hybrid compact-ENO scheme for shock–turbulence interaction problems J. Comput. Phys. 127 27–51 [11] Lele S K 1992 Compact finite difference schemes with spectral-like resolution J. Comput. Phys. 103 16–42 [12] Williamson J H 1980 Low-storage Runge–Kutta schemes J. Comput. Phys. 35 48–56 [13] Fernholz H H and Finley P J 1980 A critical commentary on mean flow data for two-dimensional compressible turbulent boundary layers AGARD-AG-253 [14] Fernholz H H, Smits A J, Dussauge J P and Finley P J 1989 A survey of measurements and measuring techniques in rapidly distorted compressible turbulent boundary layers AGARDOgraph 315 [15] Gaviglio J 1987 Reynolds analogies and experimental study of heat transfer in the supersonic boundary layer Int. J. Heat Mass Transfer 30 911–26 [16] Rubesin M W 1990 Extra compressibility terms for Favre-averaged two-equation models of inhomogeneous turbulent flows NASA CR-177556 [17] Friedrich R 1999 Modelling of turbulence in compressible flows Transition, Turbulence and Combustion Modelling (ERCOFTAC Series vol 6) ed A Hanifi et al (Dordrecht: Kluwer) pp 243–348 [18] Sarkar S 1995 The stabilizing effect of compressibility in turbulent shear flow J. Fluid Mech. 282 163–86 Journal of Turbulence 2 (2001) 001 (http://jot.iop.org/)

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increase in the Mach number increases the magnitude of the normal components of the Reynolds stress anisotropy and decreases the shear stress component. The most useful normalization of the terms in the Reynolds stress budgets is by τw um /h (which represents the energy input per unit of wall distance and was used in [4]), because all the profiles collapse onto the incompressible data in the core region. The effect of compressibility then appears in the wall region alone. It is characterized by a suppression of all production, dissipation and redistribution terms, and a vanishingly small pressure–dilatation correlation and compressible dissipation rate. Hence, the effect is clearly of a structural rather than an energetic nature. Based on these findings, the following tentative cycle of compressibility-related effects emerges. As the Mach number is increased from 0 to 1.5, the longitudinal Reynolds stress component provides less kinetic energy to the spanwise and wall-normal components. This reduces the production rate of the shear stress and hence its level. A suppressed shear stress in turn suppresses the production of the longitudinal component. Inspection of all source and sink terms in the budget of this component is finally needed to explain its increase due to compressibility.

Turbulent supersonic channel flow [19] Simone A, Coleman G N and Cambon C 1997 The effect of compressibility on turbulent shear flow: a rapid-distortion-theory and direct-numerical-simulation study J. Fluid Mech. 330 307–38 [20] Hamba F 1999 Effects of pressure fluctuations on turbulence growth in compressible homogeneous shear flow Phys. Fluids 11 1623–35 [21] Vreman A W, Sandham N D and Luo K H 1996 Compressible mixing layer growth rate and turbulence characteristics. J. Fluid Mech. 320 235–58 [22] Freund J B, Moin P and Lele S K 1997 Compressibility effects in a turbulent annular mixing layer Report TF-72, Department of Mechanical Engineering, Stanford University, Stanford, CA 94305, USA [23] Dubief Y 2000 Simulation des grandes ´echelles de la turbulence de la r´egion de proche paroi et des ´ecoulements d´ecoll´es Th`ese de Doctorat Institut National Polytechnique de Grenoble

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