Introduction on DNS

frogfish

Introduction on DNS of compressible turbulent channel flow
It is important to clarify the detailed mechanism of wall-bounded compressible
turbulent .ow for engineering and industrial applications. Since the 1950s, many
experimental studies have provided valuable knowledge about the friction coe.cient,
the mean velocity pro.les and so on (e.g. see Bradshaw 1977; Fernholz & Finley 1977,
1980; Spina, Smits & Robinson 1994; Smits & Dussauge 1996). The compressibility
e.ects are commonly divided into two types: a mean variable property e.ect due
to the variations in mean properties such as density and viscosity, and an e.ect
due to .uctuations of thermodynamic quantities, mean dilatation and its .uctuation.
Morkovin (1962) proposed the hypothesis that the compressibility e.ect was mainly
due to the variable property e.ect and that the turbulence structures of compressible
boundary layers were comparable with those of incompressible ones when the variable
property e.ect was taken into account (see Morkovin 1962; Bradshaw 1977; Smits &
Dussauge 1996). This hypothesis has long been widely acknowledged to be correct
in the study of the wall-bounded compressible turbulent .ow, and is referred to as
‘Morkovin’s hypothesis’. In the analysis of wall-bounded compressible turbulent .ow,
the Van Driest transformation (see Van Driest 1951; Rotta 1960), which is supported
by Morkovin’s hypothesis, is well known. A Reynolds analogy which relates the mass
transfer to the heat transfer is well known in incompressible turbulent shear .ow
(see White 1991). Morkovin (1962) also reported that the Reynolds analogy could
be applied to the wall-bounded compressible turbulent .ow. This concept is referred
to as the ‘Strong Reynolds Analogy (SRA)’. The SRA is applicable to the adiabatic
wall for compressible turbulent .ow. Recently, some modi.ed Reynolds analogies
applicable to the isothermal wall have been proposed by Gaviglio (1987), Rubesin
(1990) and Huang, Coleman & Bradshaw (1995).
Fernholz & Finley (1980) observed in compressible turbulent zero-pressure-gradient
boundary layer .ows on isothermal and adiabatic walls that the Van Driest
transformed velocity pro.le agreed well with the data of incompressible turbulent
.ow. On the other hand, Zhang et al. (1993) reported that the untransformed velocity
pro.le near the adiabatic wall agreed well with the data of incompressible turbulent
.ow. Huang & Coleman (1994) pointed out that the Van Driest transformation did
not work well for low Reynolds number .ow, while it was useful for high Reynolds
number .ow (see also Fernholz & Finley 1980; Spina et al. 1994). In spite of many
experimental e.orts, the mean velocity pro.le of wall-bounded compressible .ow
remains unclear. Other statistics not su.ciently understood, because experimental
measurements, such as of thermodynamic state quantities in high-speed .ow, are very
di.cult (see Spina et al. 1994; Smits & Dussauge 1996). The mean temperature pro.le
in the wall-normal direction has in practice been often estimated by using the mean
velocity in the experimental study.
In the past decade, with the rapid growth of computational resources, direct
numerical simulations (DNS) have been performed as an alternative method to
investigate the wall-bounded compressible turbulent .ow (e.g. Coleman, Kim &
Moser 1995; Guarini et al. 2000; Maeder, Adams & Kleiser 2001; Morinishi,
Tamano & Nakabayashi 2003). DNS of wall-bounded compressible turbulent .ow is
appealing because it provides three-dimensional and time-dependent data which are
very di.cult or even impossible to obtain experimentally. However, there is still a scarcity of DNS results despite their engineering importance. Typical are the DNS
results of Coleman et al. (1995) for turbulent channel .ow between isothermal walls,
using the DNS algorithm based on the Legendre Galerkin method. And Guarini
et al. (2000) performed a DNS of the boundary layer .ow on an adiabatic wall,
using the DNS algorithm based on the B-spline Galerkin method. They reported
that the Van Driest transformed velocity agreed well with the data on the wallbounded
incompressible turbulent .ow. However, the in.uence of di.erent wallboundary conditions (isothermal and adiabatic conditions) on the mean velocity and
temperature pro.les has not been investigated. Comparison of compressible and
incompressible turbulent .ows in terms of the temperature .eld in addition to the
velocity .eld is also very important for accurate understanding of the compressible
turbulent .ow. However, there have been no studies to date in which the statistics
relating to the temperature of compressible turbulent .ow are compared to those of
incompressible ones.

frogfish

In terms of turbulence statistics, Coleman et al. (1995) and Guarini et al. (2000)
reported that the variable property e.ect should be taken into account in the scaling.
However, thermodynamic .uctuations such as density and temperature have not been
examined. So, Gatski & Sommer (1998) reported that Morkovin’s hypothesis was
applicable to the near-wall asymptotic behaviour of turbulence statistics not shown
in logarithmic coordinates. Huang et al. (1995) showed that their modi.ed Reynolds
analogy agreed well with the DNS data of Coleman et al. (1995) for compressible
turbulent channel .ow between isothermal walls. Guarini et al. (2000) showed that
the modi.ed Reynolds analogy proposed by Huang et al. (1995) was e.ective for
boundary layer .ow on an adiabatic wall. However, the applicability and usefulness
for other .ows strongly a.ected by the opposite wall (e.g. the turbulent channel .ow
between adiabatic and isothermal walls) have not been examined.
A detailed understanding of the energy transfer in wall-bounded compressible
turbulent .ow requires data on the turbulent kinetic, mean kinetic and internal energy
budgets, because the energy is exchanged among them (see Lele 1994; Huang et al.
1995). However, fewer DNS data on the energy budgets for wall-bounded compressible
turbulent .ow are available than on wall-bounded incompressible turbulent .ow.
Huang et al. (1995) investigated the energy transfer near an isothermal wall using
the DNS data on compressible turbulent channel .ow presented by Coleman et al.
(1995). Guarini et al. (2000) reported that the turbulent kinetic energy budget near
an adiabatic wall in compressible turbulent boundary layer .ow was almost the
same as that of the corresponding wall-bounded incompressible .ow. However, the
di.erence and similarity between energy transfers near isothermal and adiabatic walls is still unknown. In addition, compressible and incompressible .ows have not been
compared. In order to understand wall-bounded compressible turbulent .ow, it is very
important to clarify near-wall turbulence structures in addition to mean velocity
and temperature pro.les, turbulence statistics and energy transfers. Knowledge on
the near-wall turbulence structure for incompressible turbulent .ow was summarized
by Robinson (1991). On the other hand, there have only been a few studies of
near-wall turbulence structure for compressible turbulent .ow. Coleman et al. (1995)
reported that streak structures near an isothermal wall became more coherent in the
streamwise direction as the Mach number increased. Guo & Adams (1995) performed
DNS of a compressible boundary layer .ow developing on a laminar adiabatic wall,
an adiabatic wall with constant wall temperature, and showed that streak structures
near the wall were larger than those of incompressible turbulent .ow. Wang &
Pletcher (1996) performed a large-eddy simulation (LES) of isothermal channel .ow
between hot and cold walls for almost zero Mach number, and reported that the
cold wall side exhibited stronger coherence of the near-wall streak structure. LES
of turbulent channel .ow with constant heat .ux for almost zero Mach number
was also performed by Dailey & Pletcher (1999) who showed that the turbulent
structures appeared to be more coherent on the cold wall side and less coherent on
the heating wall side. However, why the modi.cation of the near-wall streak structures
occurs is still not understood. In particular, the detailed turbulence structures near
an adiabatic wall have not been found so far. It is also uncertain whether or not
Morkovin’s hypothesis is successful relative to turbulence structures near adiabatic
and isothermal walls.
The purpose of the present study is to investigate the detailed mechanisms in
wall-bounded compressible turbulent .ow. In particular, wall-bounded .ow with an
adiabatic wall is still not understood in spite of its engineering importance. In order to
clarify compressible turbulent .ows near isothermal and adiabatic walls, we perform
DNS of compressible turbulent channel .ow between adiabatic and isothermal walls,
which has not been done previously. The present study of compressible turbulent
channel .ow between adiabatic and isothermal walls is also very important in
complementing the studies of Coleman et al. (1995) and Guarini et al. (2000).
There is still no universal theory for wall-bounded compressible turbulent .ow.
Consequently, understanding has usually been obtained from comparison with
the incompressible case, for instance the Van Driest transformation. Therefore,
understanding of wall-bounded incompressible turbulent .ow is also very important
for the wall-bounded compressible case. We focus on the similarity and di.erence
between compressible and incompressible turbulent channel .ows, as well as the
e.ects of di.erent boundary conditions on compressible turbulent .ow. To clarify the
di.erence and similarity between compressible and incompressible turbulent channel
.ows, DNS of incompressible turbulent channel .ow with passive scalar transport
between adiabatic and isothermal walls is also carried out. This .ow is one of
three speci.c situations distinguished by Teitel & Antonia (1993), and it has not
been previously studied by DNS. DNS of incompressible turbulent .ow between
isothermal walls, which corresponds to that of Kim & Moin (1989), is performed for
comparison.

frogfish

REFERENCES
Antonia, R. A., Teitel, M., Kim, J. & Browne, L. W. 1992 Low-Reynolds-number effects in a fully
developed turbulent channel flow. J. Fluid Mech. 236, 579–605.
Bradshaw, P. 1977 Compressible turbulent shear layers. Annu. Rev. Fluid Mech. 9, 33–54.
Coleman, G. N., Kim, J. & Moser, R. D. 1995 A numerical study of turbulent supersonic
isothermal-wall channel flow. J. Fluid Mech. 305, 159–183.
Dailey, L. D. & Pletcher, R. H. 1999 Large eddy simulation of constant heat flux turbulent
channel flow with property variations. AIAA Paper 98-0791.
Fernholz, H. H. & Finley, P. J. 1977 A critical complication of compressible turbulent boundary
layer data. AGARD-AG 223.
Fernholz, H. H. & Finley, P. J. 1980 A critical commentary on mean flow data for two-dimensional
compressible turbulent boundary layers. AGARD-AG 253.
Friedrich, R. & Bertolotti, F. P. 1997 Compressibility effects due to turbulent fluctuations.
Appl. Sci. Res. 57, 165–194.
Gaviglio, J. 1987 Reynolds analogies and experimental study of heat transfer in the supersonic
boundary layer. Intl J. Heat Mass Transfer 30, 911–926.
Guarini, S. E., Moser, R. D., Shariff, K. & Wray, A. 2000 Direct numerical simulation of a
supersonic turbulent boundary layer at Mach 2.5. J. Fluid Mech. 414, 1–33.
Guo, Y. & Adams, N. A. 1995 Numerical investigation of supersonic turbulent boundary layers with
high wall temperature. Proc. 5th Summer Prog. 1994, NASA/Stanford Center for Turbulence
Research, pp. 245–267.
Horiuti, K. 1992 Assessment of two-equation models of turbulent passive-scalar diffusion in channel
flow. J. Fluid Mech. 238, 405–433.
Huang, P. G. 1995 Relations between viscous diffusion and dissipation of turbulent kinetic energy.
Presented at the 10th Symp. on Turbulent Shear Flows, Penn. State U., August 14–16, pp. 2-79–
2-84.
Huang, P. G., Bradshaw, P. & Coakley, T. J. 1994 Turbulence models for compressible boundary
layers. AIAA J. 32, 735–740.
Huang, P. G. & Coleman, G. N. 1994 Van Driest transformation and compressible wall-bounded
flows. AIAA J. 32, 2110–2113.
Huang, P. G., Coleman, G. N. & Bradshaw, P. 1995 Compressible turbulent channel flows: DNS
results and modelling. J. Fluid Mech. 305, 185–218.
Kim, J. & Moin, P. 1989 Transport of passive scalars in a turbulent channel flow. Turbulent Shear
Flows 6, pp. 85–96. Springer.
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low
Reynolds number. J. Fluid Mech. 177, 133–166.
Kleiser, L. & Schumann, U. 1980 Treatment of incompressibility and boundary conditions in 3-D
numerical spectral simulations of plane channel flows. Proc. 3rd GAMM Conf. on Numerical
Methods in Fluid Mechanics (ed. E. H. Hirschel), pp. 165–173. Vieweg, Brauschweig.
Lechner, R., Sesterhenn, J. & Friedrich, R. 2001 Turbulent supersonic channel flow. J. Turbulence
2, 1–25.
Lee, M. J., Kim, J. & Moser, P. 1990 Structure of turbulence at high shear rate. J. Fluid Mech. 216,
561–583.
Lele, K. 1994 Compressibility effects on turbulence. Annu. Rev. Fluid Mech. 26, 211–254.
Maeder, T., Adams, N. A. & Kleiser, L. 2001 Direct simulation of turbulent supersonic boundary
layers by an extended temporal approach. J. Fluid Mech. 429, 187–216.
Morinishi, Y., Tamano, S. & Nakabayashi, K. 2003 A DNS algorithm using B-spline collocation
method for compressible turbulent channel flow. Computers Fluids 32, 751–776.
Morkovin, M. V. 1962 Effects of compressibility on turbulent flows. In M´e
canique de la Turbulence
(ed. A. Favre), pp. 367–380. CNRS.
Nicoud, F. C. 1999 Numerical study of a channel flow with variable properties. CTR Annual
Research Briefs 1998, Stanford Univ./NASA Ames, pp. 289–310.
Robinson, S. K. 1991 Coherent motions in the turbulent boundary layer. Annu. Rev. Fluid Mech.,
23, 601–639.
Rotta, J. C. 1960 Turbulent boundary layers with heat transfer in compressible flour. AGARD
Rep. 281. NATO.
Rubesin, M. W. 1990 Extra compressibility terms for Favre-averaged two-equation models of
inhomogeneous turbulent flows. NASA Contractor Rep. 177556.
Sarkar, S. 1995 The stabilizing effect of compressibility in turbulent shear flow. J. Fluid Mech. 282,
163–186.
Sarkar, S., Erlebacher, G., Hussaini, M. Y. & Kreiss, H. O. 1991 The analysis and modelling of
dilatational terms in compressible turbulence. J. Fluid Mech. 227, 473–493.
Smith, C. R. &Metzler, S. P. 1983 The characteristics of low-speed streaks in the near-wall region
of a turbulent boundary layer. J. Fluid Mech. 129, 27–54.
Smits, A. J. & Dussauge, J. P. 1996 Turbulent Shear Layers in Supersonic Flow. American Institute
of Physics.
So, R. M. C., Gatski, T. B. & Sommer, T. P. 1998 Morkovin hypothesis and the modeling of
wall-bounded compressible turbulent flows. AIAA J. 36, 1583–1592.
Spalding, D. B. 1961 On similarity solutions for free-convection flow past flat plates. Trans. ASME:
J. Appl. Mech. 23, 455–458.
Spina, E. F., Smits, A. J. & Robinson, S. K. 1994 The physics of supersonic turbulent boundary
layers. Annu. Rev. Fluid Mech. 26, 287–319.
Tamano, S. 2002 Direct numerical simulation of wall-bounded compressible turbulent flow. PhD
thesis, Nagoya Institute of Technology.
Tamano, S. & Morinishi, Y. 2002 DNS of wall-bounded compressible turbulent flow. Proc.
5th JSME–KSME Fluids Engng Conf., Nagoya, Japan, Novemver 17–21, pp. 1461–1466 (CDROM).
Teitel, M. & Antonia, R. A. 1993 Heat transfer in fully developed turbulent channel flow:
comparison between experiment and direct numerical simulations. Intl J. Heat Mass Transfer
36, 1701–1706.
Van Driest, E. R. 1951 Turbulent boundary layer in compressible fluids. J. Aero. Sci. 18, 145–160.
Wang, W. P. & Pletcher, R. H. 1996 On the large eddy simulation of a turbulent channel flow
with significant heat transfer. Phys. Fluids 8, 3354–3366.
Werne, J. 1995 Incompressibility and no-slip boundaries in the Chebyshev-tau approximation:
Correction to Kleiserand Schumann’s influence-matrix solution. J. Comput. Phys. 120, 260–
265
White, F. M. 1991 Viscous Fluid Flow, 2nd edn. McGraw-Hill.
Zhang, H. S., So, R. M. C., Speziale, C. G. & Lai, Y. G. 1993 Near-wall two-equation model for
compressible turbulent flows. AIAA J. 31, 196–199.