Jounml of Scienti/Tc Comput#tg. VoL 11, No. 2, 1996
An Algebraic-Q4 Turbulence Model for Attached and Separated Airfoil Flows A. Yakhot, I E. Shalman, l O. Igra, l and Y. Yadlin 2 Received March 15, 1995 An algebraic eddy viscosity model, based on a new length scale has been developed. The model proposes the eddy viscosity as a solution of a quartic (Q4) equation. The turbulent length scale for attached and separated flows is defined by employing a vorticity function F = yf2D introduced in the BaldwinLomax model. The algebraic-Q4 eddy viscosity model was incorporated into Navier-Stokes code and tested for complex transonic airfoil flows with separation. The results are compared with the experimental data. KEY WORDS:
Eddy viscosity: renormalization group: airfoil flows.
1. INTRODUCTION The problem of describing of turbulent flows has been the subject of intensive study for nearly a century. With the advent of high-speed computers in the last twenty years, hopes have been raised that modeling of turbulence would succeed as an easy-to-use, effective tool for a wide variety of engineering applications. There have been many successes, including the modeling of turbulent flows in simple geometries like jets and wallbounded flows, especially when no complicated physical phenomena like flow separation or shock waves appear in the flow field. The principles underlining the development of accurate and efficient numerical schemes for solving the compressible viscous conservation laws are reasonably well understood. However, it appears that further progress in the area of turbulent flow simulation will require a closely coupled ~Pearlstone Center for Aeronautical Engineering Studies, Department of Mechanical Engineering, Ben-Gurion University of the Negev, Beersheva 84105, Israel. -' McDonnell Douglas Aerospace, Advanced Transport Aircraft Development, Mail Stop 71-35, 1510 Hughes Way Long Beach, California 90810-1870. 71 854/11/2-1
0885-7474/96/0600-0071509.50/0 ~f~, 1996 Plenum Publishing Corporation
72
Yakhot, Shalman, Igra, and Yadlin
development of both numerical and accurate turbulence models. Progress in this area is essential for many problems of mathematical, physical, and engineering importance. To solve these problems it is necessary to develop a consistent mathematical theory that can be used to derive models from governing continuum dynamical equations. Yakhot and Orszag (1986) developed a new set of analytical tools based on renormalization group (RNG) ideas that gives a calculus for the derivation of transport approximation in turbulent flows. The simplest model suggested by the RNG theory, is an algebraic eddy viscosity model in which there is a length scale which must be specified outside the theory. The RNG-based algebraic eddy viscosity models are based on a form of the turbulent eddy viscosity suggested by the RNG approach to turbulence. The RNG theory involves systematic elimination of small scales of motion from the Navier-Stokes equations. By eliminating the small-scale modes from the wave number interval A r < k < A o , the expression for the turbulent viscosity has been derived. The RNG turbulent viscosity is expressed in terms of the mean dissipation rate ,~, and the length scale zl = 2nA/:-I, corresponding to the smallest fluctuating scales retained in the system after the RNG procedure of scale elimination is completed. In transport models of turbulence all fluctuating modes are eliminated, so A must be expressed in terms of an integral length scale (/). In terms of ~ and /, the RNG expression for the turbulent viscosity may be written in the form:
[
v=v o I+H
Ia - - T - - C )l l/3
(1.1)
where Vo is molecular viscosity, C = O(100) is the RNG constant, H(x) is the ramp function defined by H(x)=max(x, 0), and a is a constant. Due to the ramp function H(x) which appears in Eq.(1.1), the turbulent viscosity automatically turns on (v > vo) when adl4/v~ > C. This extremely simple representation of laminar-turbulent transition causes the RNG turbulent viscosity to be a peculiar one. In the algebraic models, the length scale (l) is defined as an algebraic function of position in the flow field, For modeling of a transonic turbulent flow over an airfoil, Martinelli and Yakhot (1989) successfully used the R N G turbulent viscosity expression in Eq. (1.1) and postulated a length scale in the form of a harmonic mean which is proportional to the distance from the wall in the near-wall (inner) region and which is of the order of the boundary layer thickness in the outer region. This algebraic model was analyzed by Lund (1990), who pointed out that the model contains a discontinuity in the eddy viscosity. The discontinuity appears at the point of laminar-turbulent transition which occurs for transitional boundary layers
Algebraic Turbulence Modeling
73
or in the near wall region for fully turbulent flows. When this discontinuity is removed numerically, the model yields oscillations of the eddy viscosity. Sakya et al. (1993) suggested a new length scale formulation for the RNGbased algebraic model and reported excellent agreement with experimental data for a transonic airfoil flow and for a turbulent boundary layer flow over a flat plate. Recently, the algebraic model developed in (Martinelli and Yakhot, 1989) has been modified and successfully used for the Navier-Stokes calculation of three-dimensional turbomachinery flows (Kirtley, 1992). An algebraic eddy viscosity model, based on a new length scale, which depends on boundary layer flow characteristics (displacement thickness, momentum-loss thickness) has been developed recently by Yakhot et al. (1992). The negative features pointed to by Lund (1990) have been corrected in this new formulation for the length-scale model. The model proposes the eddy viscosity as a solution of a quartic (Q4) equation. Using this new approach, numerical simulation of transitional flow in a boundary layer developed over a flat plate has been conducted, and the integral characteristics of the turbulent boundary layer have been reported to be in a very good agreement with available experimental findings. The algebraic-Q4 eddy viscosity model of Yakhot et al. (1992) has been tested for very simple geometry flows (flat plate and pipe) in which there is no separation. Recently, this model has been modified for use in complex airfoil flows with separation (Yakhot et al., 1995). In the present paper, we provide a completely self-consistent formulation of the model and emphasize the aspects related to the ability of the model to describe laminar-turbulent transitional effects. 2. ALGEBRAIC TURBULENCE MODELS
Algebraic turbulence models have their origin in boundary layer theory when experimental data accumulated during many years of an extensive research serves as a benchmark for turbulence modeling. The validity of each turbulence model is tested by its ability to predict the available experimental data. The most popular turbulence models, the eddy viscosity models, which have been widely used in engineering computations, are based on a twolayer concept wherein the boundary layer is formally split into inner and outer regions, Different turbulent length and velocity scales are used in these regions. In the algebraic turbulence models, an expression for the eddy viscosity or length scale is explicitly defined within the inner/outer region and two classes of models could be distinguished: V,,,.b = min(v/,,,,.,., v,,,,,,.,.)
(2.1)
74
Yakhot, Shalman, Igra, and Yadlin
or
I,,,,.b = min(l,,,,,.,., lo,,,c,.)
(2.2)
The crossover between the inner and outer regions is usually defined by a location where the values of the characteristic length (or the eddy viscosity) computed from the inner and outer formulations are equal.
2.1. Prandtl Mixing-Length Model
For a turbulent boundary layer flow, the Prandtl mixing length hypothesis leads to the following turbulence model v ,,,,.b = Ic21"-£2
(2.3)
¢2Q 2 / 1/2 is the vorticity, ~ 0 = ½(OUi/O.x)-OUj/Ox~), and lc is the where ~ =,___0, von Karman constant. The mixing length / is equal to the distance measured from the wall in the near wall (inner) region while in the outer part of the boundary layer / is proportional to the boundary layer thickness d. On the basis of experimental data, Patankar and Spalding (1970) suggested that
l = min(y, ),d)
(2.4)
The constant ), depends on the character of the turbulent flow in the outer region and lies between 0.18 and 0.25. For a compressible turbulent boundary layer flow over a flat plate, the constant ),=0.225 fits the available experimental data fairly well (Patankar and Spalding, 1967).
2.2. Cebeci-Smith Model (CSM) In the two-layer Cebeci-Smith model (Cebeci and Smith, 1974), the turbulent viscosity is defined differently in the inner and outer regions. In the inner region, the classical Prandtl mixing length formulation is used, viz. Vi,,,,,.,.=l~,,,,erg2,
ls,,,,,,,.=xyD
(2.5)
where the vorticity (2 is defined by ~ = (_._,)) -, 120 = ½(OUJOxj-OUJOx~), h" is the von Karman constant and D is the van Driest damping function.
Algebraic Turbulence Modeling
75
In the outer region, the CSM applies Clauser's formulation with Klebanoff's intermittency function FK/,+
v,,,,,,~ = o~U~ c~*Fm,+
(2.6)
where c~= 0.0168 and 6* is the displacement thickness. Here the displacement thickness is used to define the length scale in the outer region.
2.3. Baldwin-Lomax Model ( B L M )
Baldwin and Lomax (1978) first recast the length scale definition by finding an alternative expression patterned after the Cebeci-Smith model. In the inner region, the BLM eddy viscosity is identical to the CSM formulation in Eq. (2.4). The Baldwin-Lomax algebraic eddy viscosity model has been developed for separated turbulent flows. In the Navier-Stokes computations of such flows, computing of the displacement thickness 6* used in the CSM leads to serious numerical difficulties arised from the need for determining the boundary layer thickness and the outer edge of the layer. To avoid these difficulties, the BLM employs the vorticity function F = yt2D. The eddy viscosity in the outer region is defined by
v,,,,,.,. = ocCi F~LF~I,+
(2.7)
where 0c= 0.0168, C~ = 1.6 and FSL is defined by
FSL = min( Ym,x Fn..... 0"25ymax U~ia'/Fmax)
(2.8)
Here Fm~,.,is the maximum value of the vorticity function F that occurs in the velocity profile and Ym~., is the normal to the wall ),-location where F takes the maximum value. Udin-is the difference between maximum and minimum velocity in the velocity profile. For attached boundary layer flows, the BLM is based on the quantities Fm~x and y .... which could be interpreted as a characteristic velocity and length, respectively, and are determined from the function F = yt2D. For separated flows, turbulent characteristic length and velocity are accosiated with )'max and U~irr/F . . . . respectively. The Baldwin-Lomax algebraic model suggests different expressions for the eddy viscosity for the inner and outer layers: vi,,,,e,.~ y2g2 and Vo,,er~ )"maxF .... (or Ymax U d i f f / F m a x ) • A detailed comparison between the Cebeci-Smith and the Baldwin-Lomax models is available in (Stock and Haase, 1987).
76
Yakhot, Shalman, lgra, and Yadlin
2.4. A l g e b r a i c - Q 4 M o d e l
In this section we develop a new algebraic eddy viscosity tbrmulation based on the turbulent viscosity expression in Eq. (1,1) suggested by the RNG theory. We replace the mean dissipation rate (d) in Eq. ( 1.1 ) by the well-known expression for the eddy viscosity, used in K - - d transport models of turbulence, i.e., ~ --- K'-/v. Using this approach, the RNG expression in Eq. (1.1) may be recasted as v=vo
l+H~d~v~-C
(2.9)
In Eq. (2.9), H ( x ) is the ramp function defined by H(x) = max(x, 0), and d is a constant. From Eq. (2.9), the eddy viscosity (v) is a solution o f a quartic (Q4) equation. It should be noted that this equation possesses much more nonlinear structure because the turbulent kinetic energy (K) and the length scale (/) must be expressed in terms of mean velocity field which depends on the eddy viscosity. In the framework of the algebraic model the length scale (/) and the turbulent kinetic energy (K) must be expressed in terms of mean velocity field. One can assume two possible approximations, viz. (I) (2)
K - I20 -` f2 = (2s'2~.)'/2, Oft= ½(aUJa.x)K.,. I'-S 2 S =
9 ~ I/2 , (_STj)
agl/aXi)
S o.= ½(OU,/O.ri + @
[email protected],)
We expect that the first expression should be used for vorticity dominated flows (i.e., flow in a cavity or flow over a backward facing step), while the second one should be used in shear dominated flows. For boundary layer flows S ~ f 2 because the derivative OU/Oy is much greater than other derivatives (y is the coordinate across the boundary layer). This seems to be true even for an elongated separation region when the length of the separation bubble is much greater than its width. Therefore, for airfoil flows, considered in this paper, we employ the first approximation for K and use the following expression for the eddy viscosity v=v 0
[ ( I+H
h"s
~
C
(2.10)
I'VD
We formally introduced the a'S-coefficient into Eq. (2.10) which is possible because a length scale 1 is not yet defined. Equation (2.10) can be written in a form of a quartic equation (Q4) for the total (molecular plus turbulent) viscosity Q4(v) = v4 + ( C - 1)1'3V-- (Ka,Q/2)4 = 0,
v = max(v, vo)
(2.11)
Algebraic Turbulence Modeling
77
where C is the RNG constant (will be discussed later). The quartic equation in Eq.(2.11) has only one positive root. Indeed, Q4(0) 0 for v > 0 and Q 4 ( + c ~ ) > 0 . Due to the ramp function H ( x ) in Eq. (2.10), Eq. (2.11 ) is solved under the constraint v = max( v, vo). The derived quartic equation for the turbulent viscosity deserves some comments. An important feature of Eq. (2.11) is that the turbulent viscosity turns on (in the sense that v> 1,o) when I¢2QI2vo I > C I/4
(2.12)
for some constant C. In other words, the algebraic-Q4 equation implies the existence of a viscous sublayer where the non-dimensional number ic2£2/2v~7~, which is in fact a local Reynolds number, is less than the threshold C ~/4. Thus, the low-Reynolds number laminar-turbulent transition is built into the structure of the RNG eddy viscosity formula which alleviates the need to use explicit near-wall damping functions like in the Prandtl-van Driest formulation in Eq. (2.5). It should be noted that the turbulent viscosity formulation in Eq. (2.11) not only suggests the laminarturbulent transition but produces a critical Reynolds number which is directly related to the Kolmogorov dissipation wavenumber cutoff A,I: C = a d A , 7 4 V o 3. Another limit is the case of fully turbulent regime when I,>> vo, v ~ v,,m, and one can neglect the second term in Eq. (2.11). In this case, we obtain from Eq. (2.11) v,,,+,.~,2£212, which is the Prandtl mixing-length formulation in Eq. (2.3). 3. T U R B U L E N T L E N G T H SCALE Like other algebraic models, the model derived in the previous section, the Q4-model, requires a definition for the turbulent length scale. To define the length scale in the outer region we use the vorticity function F = y f 2 D (D is the van Driest damping function) introduced in the Baldwin-Lomax model (Baldwin and Lomax, 1978). We recall that the distance y .... is measured normal to the wall (in),-direction), where F takes the maximum value F ....... and define the length scale l appearing in Eq. (2.11 ) as / = min(y, ),y,..... )
(3.1)
From Eq. (3.1), the length scale in the inner region is defined as li, ...... = y. The length scale in the outer region is I,,,,,e,. = yy ....... where y is the intermittency coefficient which should be different for different types of flows. The problem is not only to choose a value for y but, in the NavierStokes calculation, to distinguish between the different regions of flow regimes: attached boundary layer, wake, separated flow etc., during the
78
Yakhot, Shalman, lgra, and Yadlin
calculations. In the present paper we suggest a shape factor H r = U ..... /Fn,,x as an appropriate parameter to distinguish between attached and separated flow regions. Here Um,x is the value of the velocity where F takes the maximum value. A requirement from the shape factor is that in different types of flow regions its value must differ considerably. In the following we consider different flow regimes.
3.1. Attached Boundary Layer In a turbulent boundary layer over a fiat plate, the velocity distribution may be approximated by (Monin and Yaglom, 1971)
=
96(1 )
2
g> o.15
(3.2)
where U,, the free stream velocity and u . = ( L , . / p ) 1/2 is the wall shear velocity. For the velocity distribution in Eq. (3.2), we find Y,.... = 0.53,
Fma.~ = 4.8u.,
U .... = U,.- 2.4u.
(3.3)
From Eq. (3.3), we find that for attached boundary layers
U,,,.~ = 0.2 U,__=_ 0.5 Fmax
(3.4)
U,
For a turbulent flow over a flat plate, using an experimental correlation for the skin friction coefficient (Schlichting, 1979) (
cr=2.
) - 0.0592 ~,, " - Re~/5
U,,I Re1= v0
(3.5)
gives the following approximation for the shape factor
U"~a" = 1.2Re~/l° - 0.5
F nlaX
(3.6)
For typical values of the Reynolds number, Re1, of the order 10 7 . 108, we find from Eq. (3.6) that U .... /Fmax m 5.5 + 7.0. Choosing y=0.45 for attached boundary layers and using Eqs. (3.3) and (3.1), yields 1o,,,e,.=0.2256 which is in agreement with the Prandtl mixing-length formulation in Eq. (2.4) of Patankar and Spalding (1967).
Algebraic Turbulence Modeling
79
3.2. Wake Flows We employ the expression derived by Coles (1956) for describing the wake mean velocity profiles U~
H,.
l+cos
zt
,
H,,.=~
1-
(3.7)
where H,,., is the Coles pressure gradient parameter and U,. is the velocity at the wake centerline. It is apparent from Eq. (3.7) that ym~.~=0.646&
Fm,~x= 1.82U,,,
U .... = U,.(1 -0.56H,,)
(3.8)
and for the shape factor one obtains Unlax
--=0.55
- 0.31H~,.
(3.9)
ltlax
Evaluating the pressure gradient parameter H,. from Eq. (3.7) at 0 < USUe < 0.2, an estimate for the shape factor in wake flows can be obtained by using Eqs. (3.7) and (3.9). For the present case Um~.,/Fm,., ~0.4. 3.3. Separated Flows To compute the value of U .... /Fm,x for separated flows, calculations of the flow around NACA0012 and RAE2822 airfoils using the BaldwinLomax turbulence model have been performed (Yakhot et al., 1995). In Fig. 1, we present distributions of Umax/Fmaxo n upper and lower surfaces of the airfoil. It is seen that Umax/Fm,x ~4.0 upstream of the shock wave location which is in agreement with our analysis of the attached flow. Within the separation region U ..... /Fma,,~ 1.0. The present computations, using the algebraic-Q4 eddy viscosity model, show that for the intermittency factor ?, = 0.25 in the region where Um~,.~/Fm,,x< 2.0 the shock location is accurately predicted. It is clear from these arguments that the value of the suggested shape parameter Uma.~/Fm~,x differs considerably for different flow regimes. Finally, the following values for the intermittency coefficient ), are chosen f0.25 7 = ~0.45
if if
Um~x/Fm~.~< 2.0 Umax/rmax > 2.0
(3.t0)
To conclude, the algebraic-Q4 turbulence model is defined by Eqs. (2.11 ), (3.1), (3.3), and (3.10).
80
Yakhot, Shalman, lgra, and Yadlin
8
6
I
I
!
I
!
I
I
!
I
I
!
I
I
I
!
i........................................ ,a!.. .... ~
4
I
0
I
-
I
I
.2
0
F
'
I
-
i
-
i
i
.4
-
,
