The basic tool required for the derivation of the Reynolds-averaged Navier-stokes (RANS) equations from the instantaneous Navier–Stokes equations is the Reynolds decomposi-tion. Reynolds decomposition refers to separation of the flow variable into the mean component and the fluctuating component[1]. The following rules will be useful while deriving the RANS equation. We begin by summarising the rules which govern time averages of fluctuating properties φ = Φ + ´φ and ψ = Ψ + ´ψ and their summation, In addition, div and grad are both differentiations, the above rules can be extended to a fluctuating vector quantity a=A+ ´a and its combinations with a fluctuating scalar φ= Φ + ´φ:
diva= divA; div(φa) = div(φa) = div(ΦA) + div(φ0a0);
div gradφ= div grad Φ (48)
Now, we consider the instantaneous continuity and Navier-Stokes equations in a Carte-sian co-ordinate system so that the velocity vector u has x-component u, y-component v and z-component w:
divu= 0 (49) This system of equations governs every turbulent flow, but we investigate the effects of fluctuations on the mean flow using the Reynolds decomposition in equations (49), (50), (51) and (52) and replace the flow variables u and p by the sum of a mean and fluctuating component. Thus
u=U+u0 u=U +u0 v=V +v0 w=W +w0 p=P+p0
Then the time average is taken, applying the rules stated in equations (47) and (48).
Considering the continuity equation (49), first we note thatdivu= divU. This yields the continuity equation for the mean flow:
divU= 0 (53)
A similar process is applied on the x-momentum equation (50). The time averages of the individual terms in this equation can be written as follows:
∂u Substitution of these results gives the time-average x-momentum equation
∂U
∂t + div(UU) + div(u0u0) =−1 ρ
∂P
∂x +νdiv(grad(U)) (54) Repetition of this process on equations (51) and (52) yield the time-average y- and z-momentum equations:
∂V Note that the terms (I), (II), (IV) and (V) in equations (54), (55) and (56) also appear in the instantaneous equations (50), (51) and (52), but the process of time averaging has introduced new terms (III) in the resulting time-average momentum equations [1]. This terms is product of fluctuating velocities and are associated with convective momentum transfer due to turbulent eddies. And put these terms on the right hand side of equations (54), (55) and (56) to reflect their role as additional turbulent stresses on the mean velocity components U, V and W:
∂U
The extra stress terms have been written out as follows. They result from six additional stresses among of three normal stresses
τxx=−ρu02 τyy=−ρv02 τzz =−ρw02 (60) and three shear stresses
τxy =τyx=−ρu0v0 τxz =τzx=−ρu0w0 τyz =τzy =−ρv0w0 (61) These extra turbulent stresses are called the Reynolds stresses. The normal stresses involve the respective variances of the x-, y- and z-velocity fluctuations.They are always
non-zero because they contain squared velocity fluctuations [4]. The shear stresses con-tain second moments associated with correlations between different velocity components [4]. If two fluctuations velocity components, e.g. u0 and v0, are independent random fluctuations the time average u0v0 would be zero. The equation set (53), (57), (58) and (59) is called the Reynolds-averaged Navier-stokes equations.
3.3.1 Standard κ− model
The Standardκ−(Launder and Spalding, 1974) model is the most widely used complete RANS model and it is incorporated in most commercial CFD codes [14]. In this model, the model transport equations are solved for two turbulence quantities i.e, κ and . The κ− turbulence model solves the flow based on the assumption that the rate of production and dissipation of turbulent flows are in near-balance in energy transfer [19].
We use κ and to define velocity scale ϑ and length scale ` representative of the large scale turbulence as follow:
ϑ=κ1/2 `= κ3/2
where κ is turbulent kinetic energy and is the dissipation of turbulent kinetic energy.
This is then related to the turbulent viscosity µt based on the Prandtl mixing length model,
µt=Cρϑ`=ρCµκ2
(62)
whereCµ is a dimensionless constant and ρ is density of the flow.
The governing transport equations for κand of the standard κ−model as follow,
∂(ρκ)
where term [I]denotes the rate of change ofκ or . In addition, term [II]and term[III]
display the transport of κ or by convection and diffusion respectively [1]. Last two terms describe the rate of production and destruction ofκ or respectively.
Physically, the rate of change of kinetic energy in[I]in Eq.(63) is related to the convection and diffusion of the mean motion of the flow. The diffusion term can be modelled by the gradient diffusion assumption as turbulent momentum transport is assumed to be
proportional to mean gradients of velocity. The production term, which is responsible for the transfer of energy from the mean flow to the turbulence, is counterbalanced by the interaction of the Reynolds stresses and mean velocity gradient. The destruction term deals with the dissipation of energy into heat due to viscous nature of the flow[18].
The equations contains five adjustable constants: Cµ, σκ, σ, C1 and C2. Based on extensive examination of a wide range of turbulent flows, the constant parameters used in the equations take the following values,
Cµ= 0.09; σκ = 1.00; σ = 1.30; C1= 1.44 and C2= 1.92 (65) where Prandtl numbersσκandσconnects to diffusivities ofκandto the eddy viscosity µt.
The standard κ − model has gained popularity among RANS models due to the following[8]:
• Robust formulation
• One of the earliest two-equation models, widely documented, reliable and affordable
• Lower computational overhead
• Excellent performance for many industrially relevant flows.
However, the model encounters some difficulties in:
• Fails to resolve flows with large strains such as swirling flows and curved boundary layers flow
• Poor performance in rotating flows
3.3.2 Standard κ−ω model
Wilcox (1988) developed the standard κ−ω two equation model. The standard κ−ω model is very similar in structure to the κ− model but the variable is replaced by the dissipation rate per unit kinetic energy,ω. If we use this variable the length scale is
`=√
κ/ω [4]. The eddy viscosity is given as follow,
µt=ρκ/ω (66)
The transport equations forκ andω in standardκ−ω model are
∂(ρκ)
where term [I]denotes the rate of change ofκ orω in the both Eq.(67) and Eq.(68). In addition, term[II]and term[III]display the transport ofκor ωby convection and diffu-sion respectively [4]. Terms[IV]and[V]describe the rate of production and destruction ofκ or ω respectively.
The model constants are as follow,
σκ = 2.0; σω = 2.0; γ1 = 0.553; β1 = 0.075 and β∗= 0.09 (69) The replacement with the variable ω allows better treatment in solving the flow near wall. Near to the wall, the boundary layer is affected by viscous nature of the flow. A very refined mesh is necessary to appropriately resolve the flow [8]. Although the near wall treatment of standard κ− model saves a vast amount of computer power, it is not sufficient to represent complex flow accurately. In the standardκ−ω formulation, the flow near wall is resolved directly through the integration of the ω equation. The advantage of the standardκ−ωmodel compared to the standardκ−model is that the ω equation is more robust and easier to integrate compared to the equation without the need of additional damping functions.
3.3.3 Shear-Stress Transport (SST) κ−ω model
The shear-stress transport (SST) κ−ω model was developed by Menter (1994) to ef-fectively blend the robust and accurate formulation of the κ−ω model in the near-wall region with the free-stream independence of theκ−model in the far field. To achieve this, theκ−model is converted into aκ−ω formulation [8]. The SSTκ−ω model is similar to the standardκ−ω model, but includes the following refinements:
• The standardκ−ω model and the transformedκ−model are both multiplied by a blending function and both models are added together. The blending function is designed to be one in the near-wall region, which activates the standard κ−ω model, and zero away from the surface, which activates the transformed κ− model.
• The SST model incorporates a damped cross-diffusion derivative term in the ω equation.
• The definition of the turbulent viscosity is modified to account for the transport of the turbulent shear stress.
• The modeling constants are different.
The Reynolds stress computational and theκequation are the same as in standardκ−ω model, but theequation transformed into an ω equation by substituting =κω. This yields
In the above equation all terms are same as inω equation (68) in standardκ−ω model excepted last term. The last term is called the cross-diffusion term, which arises during the=κω transformation of the diffusion term in the equation [1].
The model constants are as follow,
σκ = 1.0; σω,1 = 2.0; σω,2 = 1.17; γ2 = 0.44; β2 = 0.083 and β∗= 0.09 (71) Here, blending functions are used to achieve a smooth transition between standardκ−ω and transformed κ− models. Blending functions are introduced in the equation to modify the cross-diffusion term and are also used for model constants that take valueC1 for the originalκ−ω model and valueC2 in Menter’s transformed κ−model.
C =FcC1+ (1−F c)C2 (72) WhereFcis blending function. The functional form ofFcis chosen so that it (i) is zero at the wall (ii) tends to unity in the far field and (iii) produces a smooth transition around a distance half way between the wall and edge of the boundary layer [1].
The SSTκ−ω model more accurate and reliable for a wider class of flows like, adverse pressure gradient flows, airfoils, transonic shock waves than the standardκ−ω model.