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Unformatted text preview: Chapter 4 Turbulent ﬂows
4.1 Transition to turbulence
In the case of free ﬂows transition to turbulence occurs much earlier than in conﬁned ﬂows. In terms of Reynolds number, it is a matter of several hundred for the unbounded ﬂows around objects, and a matter of several thousand for the conﬁned ﬂows. Thus, the transition to turbulence for the case of parallel plates usually occurs at: The transition to turbulence for the case of rotating cylinders is usually measured in terms of Taylor number and occurs at the critical value of: 4.2 Turbulence Modeling Let’s consider the incompressible forms of the mass and momentum equations, (2.4), (2.24): 105 B C For the ﬂow in ducts the transition to turbulence occurs at DD ¡ ¨ A © 3 @987¤65" 420)(&$#¢ 1! 5 ' % 31! ' % " ! £
. © §¥ £¡ ¨¦¤¢ 106 CHAPTER 4. TURBULENT FLOWS where we introduced an abbreviation for pressuretodensity ratio: also use the conservative expression for the pressure term: which is true due to continuity (4.1). The solution to this equation system for high Reynolds numbers will result in a turbulent ﬂow ﬁeld. This ﬁeld is highly unsteady with a broad spectrum of eddies, which makes it difﬁcult, if not impossible to resolve it on even the most powerful computers. However, for some limited range of Reynolds numbers the solution can be obtained numerically. The technique that uses this direct approach of computing turbulence is called direct numerical simulation (DNS). The difﬁculty with this approach is that in order to reproduce all the turbulent eddies from the largest to the smallest, the simulation has to resolve the smallest space and time scales of turbulence. This in turn may require a very ﬁne grid and timeresolution. 4.2.1 LES models
In order to go beyond the Reynolds number limits of DNS another technique is employed. In this approach the grid cell sizes used are usually greater than the smallest turbulent eddies. This amounts to a spaceaveraging of the NavierStokes equation. In this case only the largest turbulent eddies are resolved, and the unresolved eddies are modeled as an addition to viscosity: where is the molecular viscosity, and is the eddy viscosity, associated with the cumulative action of unresolved eddies on the resolved large eddies. This approach to turbulence modeling is called Large eddy simulation. In the ﬁrst proposed LES model, Smagorinsky model, the turbulent viscosity is set proportional to the strainrate tensor (1.9), and the computational grid size:: where is the grid cell size, and the coefﬁcient of proportionality, Smagorinsky constant. £© ¡ © 1 ¡¢ © " £ © %£¦$ " #! 0 @) & (© 4 9 % 1 8©¡ 6 7©¡ #51 6" 4 & '© § & (© !© £ © 20 ) 31" ¡ © £ £© 1 ¢ " & £ '© § ¡¥ ¨¦ (4.1) (4.2) ©© ¤¡¢ ¨© ¨¡ ' £ § £ ¡ ¤¢ £¡
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2 4.2. TURBULENCE MODELING 107 In the Smagorinsky model the constant is considered ﬁxed, and is selected by matching the experimental data or those produced by DNS. In more sophisticated LES models is no longer considered a constant, and its value is determined in a more complex way, for example by comparing some integral measures produced from solutions obtained from spaceaveraged equations using different averaging ﬁlters [13]. The common feature of LES models is that they are based on solving for a timedependent ﬂow ﬁeld using spaceaveraged form of the NavierStokes equation. 4.2.2 RANS models
Historically, another turbulence modeling approach was used ﬁrst. This approach is based on time averaging, rather than space averaging of the NS equation. Let’s use the Reynolds assumption that any turbulent quantity, decomposed it into the time average and ﬂuctuating components:
1 ¨¦©" ¢ §¥ £ ¡ In particular, for the velocity components, we have: (4.4) (4.5) where nent, , is zero1 :
The averaging time is usually much greater than the time scale of turbulent uctuations. Since the time averaging interval does not stretch to inﬁnity, the resultant average quantities can still be slowly varying functions of time. But the time scale of these variations will be larger than largest turbulence time scales. Under this assumptions decomposition (4.6) is called the Reynolds decomposition.
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¡ £ 1 ¨¦¤" §¥ £ 0) , can be " & ! £ 1 ¨¦©" §¥ £ 1 ©" ¥ £ 108 Applying (4.4) to (4.1), (4.2), we have: CHAPTER 4. TURBULENT FLOWS (4.7) and after and time averaging the latter and the continuity relation (4.1), and using (4.6), we obtain: (4.8) As can be seen, both the mean and ﬂuctuating ﬁelds satisfy the continuity. The last equation is called the Reynolds averaged NavierStokes equation (RANS), thus the name of the approach. As can be seen the equation contains and extra unknown term composed of derivatives of the socalled the Reynolds stress . tensor: To close the new equation system one needs to formulate extra equations for the components of Reynolds stress tensor (closure). One of the simplest and ﬁrst closure was suggested by Boussinesq, and is called the Boussinesq approximation. In this approximation the Reynolds stress tensor is considered proportional to the mean velocity gradient: (4.9) (4.10) (4.11) where we introduced the mean strain rate tensor, analogous to (1.9). Quantity deﬁned by (4.10) is called the Turbulent kinetic energy, and is called the eddy viscosity,. Both and represent two new unknown variables in the model. Empirical algebraic relations can be devised for and connecting them to and lengthscales of the problem, as is done in the mixinglength theory [2], or in a more sophisticated RANS models discussed below. On the other end of the turbulence closure spectrum is the Reynolds stress model (RSM). In this model each component of the Reynolds stress tensor is & #© © £© 1 ( $¡ ¡§ & (© ¥" © © © ¤¡ £ £ £ § 1 ¡ ¨ £¡ £§ ¥© 7© ¡ ¡ ¤# £¡ ¡ ¤¡ ¥ £ § ¨ £¡ ¡¢ ¢ ¡ & © B ' 7©¡ ©B§ ¨ 8©¡ £ 1 ¡ 5© ¥ B "' £
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© ¤¡ £ ¥ ¥ ¦¤ 4.2. TURBULENCE MODELING 109 obtained from a separate equation. Since there are six independent components, there should be six equations. In the full Reynolds stress model these equations are represented by the PDEs of the transport type. In an approximate version of RSM  algebraic Reynolds stress model these equations are given by algebraic relations. To obtain the equations for Reynolds stress tensor, let’s ﬁrst subtract (4.8) from (4.7): which, using continuity (4.1), can be rewritten as: Let’s multiply this equation by Now add the two last equations: Now apply Reynolds decomposition (4.4): equation: , and time average the last © £© § ¢¨© § ¢ ¡ £¡ © ¡ ¡ £¡ £ 1 £© ¦ ¢ § © © ¤£( ¢ " © § ¢ ' ¡ ¢ ' £ £© 1 © ¦ ' © ¦ " ¢ § £© 1 © # ' © # " ¢ § ¡ ¡ ¡ £© ¥ © ¢ § © ¤¡ ¥ © ¢ § © ¥ £© § ¢ § © ¥ © ¤( ¢ § ¡ £ ¡ £¡ ¡ §¡ §¥ ¢ ¨#¥ £ ' £ £© 1 © § ' ¡§ ( ¨¡ ¥£ ¡ ¦ © § " ¢ § £© ¥ © ¢ § © ¡ ¡ ¥ If we swap the indexes and of this equation, we get: © © %# ¢¨© ¨¡ 5¢ £¡ § £ ¡ ' C£ £© 1 © # ' © # " ¢ § © ¤¡ ¥ © £ § © ¥ © %# ¢ ¨(¥ ¢ ¡ £ £¡ § ¡ ¡ (4.12) :
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© ( " § © ¤¡ ¥ © ¡ £ " § £© 1 © ¥ ( " ¨(¥ ¡ §¡ § © ¥ © ¤( $#¥ £¡ § ¡ ¤ ¥ ¡ £© § ¢ § ¦¥ ¦ ¡ © ¡ ¢¤¦ © ¡ ¨¤¦ This is the equation for the components of the Reynolds stress tensor. Since the Reynolds stress tensor is symmetric, it has only six independent components, and the number of equations is 6. As can be seen the equation contains 3rd order correlations: . One can write an equation for as well, but it will depend on 4th order correlations, and so on. In practice, an empirical relationship is proposed, that links the third order tensor to the second order tensor . This relationship is called closure. In fact all the terms on the RHS © ¡ ¢ ¦ © ¡ ¢ ¦ 1 © £© § ( § © ¤( § ¡ © £¡ ¡ ¡ " © § £© 1 ¦ © ( " § £ ( £© ¥ © ( § © ¤¡ ¥ © § § £© 1 ¦ # " © ¥ § §¥ ( ¡ £ ¡ ¡ ' ¡ § £ '£ ¡ © ¡ £¡ £ ' 1 © £© ¦ #§ © ¤£# ¦ " © § ( ' ¡ § C£ £© 1 © ¦ " ( § £© 1 © ¤( " § § ¡ ¡ £© ¥ © ( § © ¤¡ ¥ © § § £© 1 § ( " © ¥ § §¥ ( ¡ £ ¡ ¡ changing the order of time differentiation and time averaging, and canceling terms we obtain: § § ¥ " " § £© 1 © § © ' 1 ¡§ ©¡ 1 © £© § 1 ( $¡ ¥ " § © ¤£# 1 £© 1 © ¦ " ( ' £© 1 © § " # ¡ ¡ £© 1 © # " § ' £© 1 © ¤( " ¦ ¡ ¡ £© ¥ © ( § © ¤¡ ¥ © § § ¡ £ ¡ §¡ ¦¥ #$#¥ ¦ " § £© 1 © ¤( ' © ( " ¡ ¡ £© 1 ¦ # " © ¥ § ¡ § 1 ( ¨¡ £¡§ © § " ¡ ¥ § ¥" ' ¡ 1 ¦§ £ ¥ " C£ ' ¥ § © ¡ § ©¡ £ ¡ £ 1 £© § 1 ( $¡ ¥ " § © ¤£# 1 § § ¥ " " © § 1 #§ ¡ ¥ " ' ¡ 1 ¦§ ¥ " C£ ' £© 1 © § ' © ¦ " 1 #¨¡ ¥ " § £© 1 © ¤( ' © ( " 1 § § ¥ " § ¡ § ¡ ¡ £© ¥ © ¡ § ¡ § ¡ 1 #¨¡ ¥ " § © ¤£¡ ¥ © 1 ¦§ ¥ " § © ¥ £© § 1 #$¡ ¥ " § © ¥ © ¤£( 1 ¦§ ¡ § §¥ 1 #$¡ ¥ " ¨(¥ 1 ¦§ ¥ " §¡ ¥"
§ (4.14) and ﬁnally: (4.13) 110 which after taking into account (4.6) simpliﬁes to: CHAPTER 4. TURBULENT FLOWS 4.2. TURBULENCE MODELING 111 of (4.14) require some kind of closure to make the problem complete. If such closures are established, the obtained equation system will constitute a turbulence model. In this particular case, when the equation system provides PDEs for Reynolds stress tensor components, the corresponding turbulence model is called Reynolds stress model (RSM). Considering that the Reynolds stress tensor is symmetric, this model may include as many as 12 PDEs: 3  velocity, 1  pressure, 6  Reynolds stress tensor. The 12th equation is the one for the turbulent dissipation rate, which will be discussed in the next section. Two equation turbulence models The RSM model described above may be prohibitively expensive in terms of the number of equations and the complexity of implementation. In this case simpler models can be used, which are based on smaller number of equations. The ﬁrst step to reduce the number of equations is to relate the components of the Reynolds tensor to the mean velocity gradients following the Boussinesq approximation (4.9). Then one can formulate a separate transport equation for and an algebraic relation for as a function of the latter and the mean velocity gradients. This approach will constitute a one equation turbulence model. A more popular approach is to formulate two transport equations: one for turbulent kinetic energy, and another for it’s dissipation rate . The eddy viscosity, is then related to and as which can be shown from dimensional reasoning. Indeed, if represents the rate of change of : then its physical dimensions should be , Considering that , and , we can obtain (4.15). This approach is called the turbulence model (KE) [14, 15], which is the most popular turbulence model for engineering computations today. The equation of is formulated as follows: the where is the effective eddy viscosity of . This equation is solved with the boundary conditions of zero at the walls. Usually a nonzero is set at the § (4.16) 3 64 ££ ¤86 ¢ £ ¢ 71 ¡ 5© ¥ '£ § © %¡ £ 64 ¥" ££ © %¡ £ (4.15) 4 ¥© & ¡ ¢) £ '© & ¢ '© & § ¨ £¡ 1 £¡ 46 4 ¥ ¦© " £ ' ££ & '©
¢ ¥ ¦© & '© 112 CHAPTER 4. TURBULENT FLOWS open boundaries, where its value is related to level of turbulent ﬂuctuations expected in each particular ﬂow case. The equation for the turbulence dissipation rate, , is written in analogy to the one for . Namely, we multiply the equation by , and introduce the effective eddy viscosity of : to obtain: where we introduced two new constants . Boundary conditions on can be obtained from those on , by relating to through equation (4.16) where the nonsteady term should be set to zero at the boundary: . (diffusivNow equations (4.16) and (4.17) have three effective viscosities: ity of momentum), (diffusivity of turbulent kinetic energy, ), and (diffusivity of turbulence dissipation rate, ). It should be noted that unlike the molecular viscosity, , which is a property of the ﬂuid and is usually independent on coordinates or time2 , all turbulent viscosities are functions of space, and are therefore new dependent variables of the problem along with and . Another assumption of the model is that with different proportionality constants: ¤ ¢© &© £ ¥¤ © £ ¥© and are both proportional to (4.18) (4.19) where model. , ¤ are the effective ”Prandtl numbers”, which are constants of the The system of equations (4.16), (4.17), (4.18), (4.19), (4.15) is closed and constitutes the turbulence model. It has 5 empirical constants: the values of which are determined by comparison of computations with experimental data. KE model belongs to the class of twoequation turbulence models. Another model [16], where instead important two equation turbulence model is the of the turbulence dissipation rate a turbulence frequency scale, , is used as an extra variable.
2 In ﬂows with heat transport this may not be the case & #© £© § ¨ ¥ £ £ 5 ¤ § ©¥ ¦ ¨ ¢© ¥ £¥ ¡ 5 )' ¥ ¤ ¥ ¥© © 4 B ) ¥ §¥ ¨ % ¢) & £ 4 ) © ¥ ¨¨ £ 0) % ©§ ¥ ¦¨ ¦ ¥© (4.17) 4 4 ) 71 ¡ 5£© ¥ § © ¤£¡ ¥ " © ¤£¡ ¥ & © 0) ¨ £¡ 1 £¡ ¢© " £ § ' %§ ¡© ' £¡ &¢) © ¤ & © ...
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 Spring '11
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