Scalar Transport. Introduction. T. J. Craft George Begg Building, C41. Eddy-Diffusivity Modelling. TPFE MSc Advanced Turbulence Modelling

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1 School of Mechanical Aerospace and Civil Engineering TPFE MSc Advanced Turbulence Modelling Scalar Transport T. J. Craft George Begg Building, C41 Reading: S. Pope, Turbulent Flows D. Wilco, Turbulence Modelling for CFD Closure Strategies for Turbulent and Transitional Flows, (Eds. B.E. Launder, N.D. Sandham) Notes: Blacboard and CFD/TM web server: - People - T. Craft - Online Teaching Material Scalar Transport 2011/12 1 / 18 Introduction In man CFD applications we ma be interested in predicting the behaviour of scalar quantities such as enthalp, temperature, mass fraction of a chemical species, etc. Consider a propert Φ transported b the flow. We assume that its evolution can be described b the equation Φ ( t + ) (Ũ Φ = λ Φ ) + S Φ (1) where S Φ represents an source or sin term that ma be present. If Φ is enthalp or temperature, the flow is low speed and the effects of viscous heating are negligible, then S Φ can often be ignored. If Φ represents the mass fraction of a species being consumed or produced b chemical reaction, then S Φ ma be of great importance and has to be modelled from the details of the reaction process. If Φ represents temperature, then λ is simpl ν/pr where Pr is the molecular Prandtl number of the fluid. Scalar Transport 2011/12 2 / 18 Equation (1) can be averaged in the same wa as the momentum equations. First, we split Φ into mean and fluctuating parts: Φ(,t) = Φ() + φ(,t) (2) Substituting into equation (1) and averaging leads to an equation of the form Φ t + ( U Φ ) = ( λ Φ ) u φ + S Φ (3) where S Φ represents the averaged source terms from the original instantaneous equation. Comparing with the Renolds averaged Navier-Stoes equations, in this case the additional terms involve the turbulent scalar flues, u φ. In order to close the equation, we need to devise models for these turbulent scalar flues. For simplicit, we consider the problem of modelling turbulent heat flues, u i θ, although the principles are easil etended to other scalars. Scalar Transport 2011/12 3 / 18 Ed-Diffusivit Modelling A linear ed-viscosit model approimates the turbulent stresses b: ( Ui u i u = (2/3)δ i ν t + U ) (4) i An obvious etension of this is to introduce an ed-diffusivit for the scalar flues, related to the turbulent viscosit: u i θ = ν t σ t i (5) The total flu terms in the mean temperature transport equation then become [( ν Pr + ν ) ] t σ t The turbulent Prandtl number, σ t, is often assumed to be constant. In a near-wall flow, σ t = 0.9 is usuall adopted. However, in free flows a lower value (around 0.7) is more appropriate. Scalar Transport 2011/12 4 / 18

2 Scalar Flues in Simple Shear An eact equation can be derived for u i θ, which shows it has a generation term of the form P iθ = u i u u θ U i In simple shear flow U = U(), Θ = Θ(), this gives: Homogeneous Shear Flow U() Θ() q U() T() q Heated Plane Channel DNS: Horiuti (1992) P 1θ = uv dθ P 2θ = v 2 dθ vθ du U() Θ() Epts: Tavoularis & Corrsin (1981) Heated Aismmetric Jet This suggests both the wall-normal and wall-parallel scalar flues will be non-zero. Measurements and DNS data show that generall uθ > vθ, particularl in the near-wall region. Epts: Chevra & Tutu (1978) Scalar Transport 2011/12 5 / 18 Scalar Transport 2011/12 6 / 18 However, the ed diffusivit model gives: vθ = ν t σ t dθ and uθ = 0 ie. zero turbulent scalar flu in the streamwise direction. Maing the usual boundar laer approimations, the (stea) mean scalar transport equation becomes U + V = ( α ) vθ Consequentl, if streamwise gradients are small compared to cross-stream ones, the above error in uθ ma not have serious consequences. However, the error ma be more serious in buoanc-influenced or other comple flows. Generalized Gradient Diffusion Modelling A somewhat better model for the scalar flues arises from the Dal & Harlow (1970) Generalised Gradient Diffusion Hpothesis (GGDH): In a simple shear flow this gives vθ = c θ ε v 2 dθ u i θ = c θ ε u iu and dθ uθ = c θ uv ε This does at least give a non-zero uθ, and also better reflects the dependence of vθ on v 2, seen in the generation terms. If a stress transport model or non-linear EVM is used which gives good stress anisotrop levels, the above form can be used as is (tpicall with a coefficient c θ around 0.3). Scalar Transport 2011/12 7 / 18 Scalar Transport 2011/12 8 / 18

3 With an EVM (which will not generall provide good normal stress predictions), the GGDH can still be used with some modification. Ince & Launder (1989) used this model in conunction with a linear ed-viscosit model for the stresses, taing c θ = (3/2)(c µ /σ t ). In a shear flow (where the model returns v 2 = (2/3)) this gives vθ = c θ ε v 2 dθ = c µ 2 dθ σ t ε dθ uθ = c θ uv ε which gives vθ as in the ed-diffusivit model, but should give a better approimation of uθ. More comple algebraic scalar flu models have been proposed, some developed along similar routes to the non-linear ed-viscosit models eamined earlier for the Renolds stresses. Buoanc Effects One important application where temperature (or concentration) differences must be properl accounted for is in buoanc-affected flows. In that case, there is an additional source term in the momentum equation: D(ρU i ) =... + ρg i (6) This results in an additional term in the transport equation for the fluctuating velocit: Du i =... + ρ g i /ρ (7) where ρ is the fluctuating densit. Consequentl, one gets an additional generation term in the turbulent inetic energ equation: G = (1/ρ)g i u i ρ (8) Scalar Transport 2011/12 9 / 18 Scalar Transport 2011/12 10 / 18 Densit fluctuations are often epressed in terms of temperature fluctuations, b introducing the thermal epansion coefficient: β = 1 ρ ρ The buoanc source term in the equation then becomes (9) G = β g i u i θ (10) Note that the buoant generation of depends on the turbulent scalar flues. Depending on the sign of vθ, buoanc can either enhance or reduce levels: In a stabl stratified laer (with dθ/ > 0) we would tpicall have vθ < 0 and hence (with g 2 < 0), G is negative. In an unstable laer, dθ/ < 0, and hence vθ > 0 and G is positive. Buoanc in Stress Transport Models The buoanc-related term appearing in the transport equation for u i leads to an additional generation term in the stress transport equations: where Du i u = P i + G i + φ i ε i + d i (11) G i = β g i u θ β g u i θ (12) When modelling the pressure-strain redistribution process, a contribution due to buoanc should also be included: φ i = φ i1 + φ i2 + φ i3 (13) Within the framewor of the modelling adopted earlier, φ i3 is approimated in a similar manner to φ i2 : φ i3 = c 3 (G i (1/3)G δ i ) (14) φ i3 is thus assumed to redistribute the buoanc generation. Scalar Transport 2011/12 11 / 18 Scalar Transport 2011/12 12 / 18

4 In some instances it ma be sufficient to model the scalar flues u i θ using a GGDH approach (or similar) as described earlier. However, in strongl buoant flows this ma not be adequate, since the scalar flues are themselves affected b buoanc. Some etended algebraic heat flu models have been proposed, incorporating some buoanc effects (eg. Hanalić et al, 1996). Another option is to adopt a full second-moment closure, solving transport equations for the scalar flues also. Scalar flu transport equations can be derived in a similar manner to those for the Renolds stresses. The result can be written in the form Du i θ The generation terms P iθ and G iθ are eact: = P iθ + G iθ + φ iθ ε iθ + d iθ (15) P iθ = u i u u θ U i (16) G iθ = β g i θ 2 (17) Scalar Transport 2011/12 13 / 18 Other terms in the transport equation have to be modelled. Similar assumptions are tpicall made for these as for the corresponding terms in the stress transport equations. However, the details will not be covered in this course. The scalar flu buoanc generation depends on the scalar variance, θ 2. This can be obtained b solving its transport equation, of the form Dθ 2 where the generation rate is given b = P θ 2ε θ + diffusion (18) P θ = 2u i θ i The dissipation rate ε θ is usuall either modelled algebraicall b assuming that thermal and namic timescales are related (eg. R = (/ε)(2ε θ /θ 2 ) being constant), or obtained from its own modelled transport equation. Scalar Transport 2011/12 14 / 18 Negativel-Buoant Jet Opposed/Buoant Wall Jet Isothermal and buoant cases studied. LES data from Addad et al (2004). Aismmetric downward directed buoant et. Requires a good outer flow and near-wall modelling for accurac. Eperiment of Cresswell et al (1989). Velocit and shear stress profiles one diameter downstream of et discharge. TCL RSM; Basic RSM Vertical velocit contours. Scalar Transport 2011/12 15 / 18 Scalar Transport 2011/12 16 / 18

5 Stabl Stratified Miing Laer Studied eperimentall b Uittenbogaard (1998). U 1 = 0.5m/s, ρ 1 = 1015g/m 3 U 2 = 0.3m/s, ρ 2 = 1030g/m 3 Linear -ε scheme overpredicts turbulence levels and hence miing. Second-moment closures do better at capturing buoanc effects on scalar flues and hence reduce miing. In this case, as turbulence levels decrease, modelling of diffusion becomes more influential. References Addad, Y., Benhamadouche, S., Laurence, D., (2004) The negativel buoant et: LES results, Int. J. Heat and Fluid Flow, vol. 25, pp Chevra, R., Tutu, N.K., (1978) Intermittenc and preferential transport of heat in a round et, J. Fluid Mech., vol. 88, p Cresswell, R., Haroutunian, V., Ince, N.Z., Launder, B.E., Szczepura, R.T., (1989) Measurement and modelling of buoanc-modified elliptic turbulent shear flows, Proc. 7th Turbulent Shear Flows Smposium, Stanford Universit. Dal, B.J., Harlow, F.H., (1970) Transport equations in turbulence, Phs. Fluids, vol. 13, pp Hanalić, K., Keneres, S., Durst, F., (1996) Natural convection in partitioned two-dimensional enclosures at higher Raleigh numbers, Int. J. Heat Mass Transfer, vol. 39, pp Horiuti, K., (1992) Assessment of two-equation models of turbulent passive-scalar diffusion in channel flow, J. Fluid Mech., vol. 238, pp Ince, N.Z., Launder, B.E., (1989) On the computation of buoanc-driven turbulent flow in rectangular enclosures, Int. J. Heat Fluid Flow, vol. 10, pp Tavoularis, S., Corrsin, S., (1981) Eperiments in nearl homogeneous turbulent shear flow with a uniform mean temperature gradient. Part 1., J. Fluid Mech., vol. 104, pp Uittenbogaard, R.E., (1998) Measurement of turbulence flues in a stea, stratified, miing laer, Proc. 3rd Int. Smposium on Refined Flow Modelling and Turbulence Measurements, Too. Scalar Transport 2011/12 17 / 18 Scalar Transport 2011/12 18 / 18