STRESS TRANSPORT MODELLING 2

STRESS TRANSPORT MODELLING 2 T.J. Craft Department of Mechanical, Aerospace & Manufacturing Engineering UMIST, Manchester, UK STRESS TRANSPORT MODELLING 2 p.1

Introduction In the previous lecture we introduced the stress transport equations, and some relatively simple models for closing them. We noted that the generation terms are represented exactly, and that in many flows the pressure-correlations are then the most important terms that require modelling. The (linear) models developed for the pressure correlations were based on simple return-to-isotropy ideas. They required additional corrections (involving wall-distances) to account for wall-proximity effects. In this lecture we look at more advanced approaches. In particular, how models can be built to ensure they satisfy the 2-component turbulence limit. STRESS TRANSPORT MODELLING 2 p.2

2-Component Turbulence 2-component turbulence can be found at fluid-fluid interfaces. Velocity fluctuations normal to surface vanish, but those parallel to it need not. Consequently, v 2 = 0, but u 2 and w 2 are non-zero. Fluid 2 Fluid 1 y x Near-wall turbulence is close to 2-component. Buoyancy/rotation effects can drive turbulence towards 2- component limit. Channel flow DNS of Kim et al (1987) Although many flows not close to 2-component limit, a TCL model should have better physics built into it, and might be hoped to perform better than simpler models even away from this limit. STRESS TRANSPORT MODELLING 2 p.3

Realizability Realizability (Schumann 1977, Lumley 1978): ensures model cannot predict unphysical values eg. negative normal stresses. Applied by ensuring model behaves correctly when one stress vanishes ie. in 2-component limit. At RSM level one can control the rate of change of stresses in the 2-component limit. A necessary condition for realizability is that Dv 2 /Dt = 0 when v 2 = 0: to prevent v 2 being driven to negative values. Shih & Lumley (1985) developed realizable models for φ ij, φ iθ in the second-moment equations. Modelling at UMIST has followed similar lines, but used slightly different constraints for reasons given later. This lecture describes methods of including 2-component limit in stress transport modelling, illustrated by model development done at UMIST. STRESS TRANSPORT MODELLING 2 p.4

Stress Transport Equations Transport equation for u i u j can be written: Du i u j Dt = P ij + φ ij + d ij ε ij Production P ij = u i u k U j x k u j u k U i x k Diffusion typically modelled by simple GGDH: d ij = x k [( ) ] k νδ lk + c s ε u ui u j ku l x l Modelling of pressure-strain (φ ij ) and dissipation (ε ij ) will be considered in this lecture. STRESS TRANSPORT MODELLING 2 p.5

Wall/Surface Asymptotes One can examine the immediate near-wall behaviour by writing Taylor series expansions: u 1 = a 1 + b 1 y + c 1 y 2 + u 2 = a 2 + b 2 y + c 2 y 2 + u 3 = a 3 + b 3 y + c 3 y 2 + At solid wall, a 1 = a 2 = a 3 = 0. Furthermore, continuity leads to b 2 = 0. This leads to u 2 1, u2 3 y2, but u 2 2 y4. The near free-surface behaviour and limiting forms of all terms in the transport equations can be examined in a similar fashion. STRESS TRANSPORT MODELLING 2 p.6

Wall Asymptotes At a solid wall, for example, one can show that: d ν ij φ ij + d p ij ε ij u 2 2νb 2 1 + 12νb 1c 1 y + 4νb 1 c 1 y + 2νb 2 1 + 8νb 1c 1 y + v 2 12νc 2 2 y2 + 4νc 2 2 y2 + 8νc 2 2 y2 + w 2 2νb 2 3 + 12νb 3c 3 y + 4νb 3 c 3 y + 2νb 2 3 + 8νb 3c 3 y + uv 6νb 1 c 2 y 2 + 2νb 1 c 2 y + 4νb 1 c 2 y + STRESS TRANSPORT MODELLING 2 p.7

Anisotropy Invariants Near the 2-component state, turbulence has different characteristics than in regions close to isotropy. Anisotropy invariants can be used to give some information on turbulence structure: A 2 = a ij a ij A 3 = a ij a jk a ki a ij = u i u j /k (2/3)δ ij Realizable states lie within a triangle in A 2 -A 3 space. Lumley s flatness parameter A = 1 9/8(A 2 A 3 ). In isotropic turbulence A = 1 In 2-component turbulence A = 0 STRESS TRANSPORT MODELLING 2 p.8

Constructing TCL Models One approach might be to use A (or other invariants) to make coefficients in the linear models vanish in the 2-component limit. But one cannot get different limiting behaviour for different components this way. Alternatively, we can construct higher-order tensor forms of models, and choose the resulting model coefficients in order to give desired behaviour. The pressure-strain is often the most influential modelled term: φ ij p ρ ( ui + u ) j x j x i STRESS TRANSPORT MODELLING 2 p.9

Modelling φ ij As outlined earlier, one can obtain a Poisson equation for fluctuating pressure p: 1 ρ p x 2 i = 2 x l x k (u l u k u l u k ) 2 U k x l u l x k Solving this in terms of a Greens function gives an expression for φ ij : φ ij = 1 ( ) 3 u k u l u i + 3 u k u l u j dv 4π V ol r l r k r j r l r k r i r }{{} φ ij1 1 ( ) 2 u l u i + 2 u l u j U k dv 2π V ol r k r j r k r i r l r }{{} φ ij2 STRESS TRANSPORT MODELLING 2 p.10

Modelling φ ij2 Assuming the velocity gradient varies reasonably slowly, one can model φ ij2 as φ ij2 = ( ) Xkj li + X lj Uk ki where Xkj li = 1 x l 2π V ol 2 u l u i r k r j dv r The tensor X li kj can then be modelled in terms of the Reynolds stresses. A general linear tensor form (with required symmetries) is: X li kj = αu l u i δ kj + β (u i u j δ lk + u l u j δ ik + u l u k δ ij + u i u k δ lj ) +γu k u j δ il + ξk (δ lj δ ik + δ lk δ ij ) + ηkδ il δ kj From the exact integral, we want to ensure that: Continuity: Xki li = 0, li Normalization: Xkk = 2u lu i STRESS TRANSPORT MODELLING 2 p.11

A Linear φ ij2 Model Applying these constraints leads to (Launder et al 1975) φ ij2 = γ + 8 11 (P ij (1/3)δ ij P kk ) 30γ 2 ( Ui k + U ) j 55 x j x i 8γ 2 11 (D ij (1/3)δ ij D kk ) where D ij = u i u k U k / x j u j u k U k / x i. However, one cannot satisfy 2-component limit for any value of the free coefficient γ. STRESS TRANSPORT MODELLING 2 p.12

Non-Linear φ ij2 Modelling 1 The two-point correlations in integral for Xkj li functions of the stresses. One might thus include higher order terms in X li kj : are not simple linear X li kj k = λ 1δ li δ kj + λ 2 (δ lj δ ki + δ lk δ ij ) + λ 3 a li δ kj + + λ 10 (a lm a mj δ ki + a lm a mk δ ij + a im a mj δ lk + a km a mi δ lj ) + + λ 13 a li a km a mj + λ 14 a kj a lm a mi + + λ 20 a mn a mn (a lj δ ki + a lk δ ij + a ij δ lk + a ki δ lj ) Inclusion of cubic terms allows the 2-component limit to be satisfied: φ 222 = 2Xk2 l2 U k = 0 when u 2 2 x = 0 l Application of this, together with continuity and normalization, yields a system of equations for the λ s, which when solved gives a model with 2 free coefficients. STRESS TRANSPORT MODELLING 2 p.13

Non-Linear φ ij2 Modelling 2 The resulting model can be written: φ ij2 = 0.6 (P ij (1/3)δ ij P kk ) + 0.3a ij P kk [ [ uk u j u l u i Uk 0.2 + U ] l u lu k k x l x k k [ ]] U j U i u i u k + u j u k x l x l c 2 [A 2 (P ij D ij ) + 3a mi a nj (P mn D mn )] {( 7 + c 2 15 A ) 2 (P ij δ ij P kk /3) + 0.1 [a ij 0.5 (a ik a kj δ ij A 2 /3)] P kk 4 [( ui u m 0.05a ij a lk P kl + 0.1 k P mj + u ) ] ju m k P u l u m mi (2/3)δ ij k P ml [ ] [ [ ul u i u k u j u l u m u k u m Ul + 0.1 k 2 (1/3)δ ij k 2 6D lk + 13k + U ]] k x k x l + 0.2 u } lu i u k u j k 2 (D lk P lk ) Fu (1988) set c 2 = 0. Later work at UMIST has included non-zero values for both c 2 and c 2, to improve wall-bounded flows. STRESS TRANSPORT MODELLING 2 p.14

Modelling φ ij1 Including non-linear terms in Rotta s (1951) return to isotropy model: φ ij1 = c 1 εa ij c 1ε(a ik a kj A 2 δ ij /3) c 1εA 2 a ij Inclusion of a cubic term does not increase generality since the Cayley Hamilton theorem gives (a ik a kl a lj A 3 δ ij /3) = A 2 a ij /2. One can apply the 2-component limit behaviour in a similar manner to φ ij2 : finding coefficients that give φ 221 = 0 when u 2 2 = 0. However, the resultant model does not perform well in shear flows. An alternative is to make the coefficients functions of A, to ensure they vanish in the 2-component limit. The current UMIST form is: φ ij1 = c 1 ε [a ij + c 1(a ij a jk A 2 δ ij /2)] f Aεa ij with c 1 = 3.1(A 2 A) 1/2, c 1 = 1.1 and f A = A1/2. STRESS TRANSPORT MODELLING 2 p.15

Dissipation Modelling 1 In high-re-number flow, ε ij is normally assumed isotropic: ε ij = (2/3)εδ ij However, this form does not satisfy the 2-component limit (at a wall ε 22 = 0). A form ε ij = εu i u j /k does vanish if u 2 2 = 0. One can combine these two forms to give with f ε a suitable function of A. This does give ε 22 = 0 when u 2 2 = 0. ε ij = (2/3)εδ ij f ε + u iu j k ε(1 f ε) But note that it still does not satisfy the exact wall or surface-limiting behaviour for all components. STRESS TRANSPORT MODELLING 2 p.16

Dissipation Rate Modelling 2 The dissipation rate ε is typically obtained from its own separate transport equation: Dε Dt = c εp kk ε1 2k c ε2 ε 2 k + x k [( ) ] k ε νδ lk + c ε ε u ku l x l Anisotropy invariants can be introduced into the coefficients to make the model respond to different types of turbulence structure. The current UMIST form is: c ε2 = 1.92/(1 + 0.7A 1/2 2 A) c ε1 = 1.0 STRESS TRANSPORT MODELLING 2 p.17

Scalar Flux Modelling In some flows it is sufficient to employ a simple gradient-diffusion type of model for the scalar fluxes. In others one might need to employ a full second-moment closure. A similar treatment can be applied to the scalar flux equations to ensure realizability. The transport equation for u i θ can be written: Du i θ Dt = P iθ + φ iθ + d iθ ε iθ Production is now P iθ = u i u k T x k u k θ U i x k Diffusion can often conveniently be modelled via the GGDH. The isotropic dissipation assumption gives ε iθ = 0. The only term remaining is the pressure-scalar gradient correlation φ iθ. STRESS TRANSPORT MODELLING 2 p.18

Pressure-Scalar Gradient 1 In similar fashion to φ ij, we can obtain φ iθ = 1 3 u k u l θ dv 1 2 u l θ U k dv 4π V ol r l r k r i r 2π V ol r k r i r l r }{{}}{{} φ iθ1 φ iθ2 The rapid part, φ iθ2, can be modelled as φ iθ2 = 2b l ki U k x l where b l ki = 1 4π V ol 2 u l θ r k r i dv r Linearity principle: φ iθ is linear in the scalar, so b l ki stresses, but only linear in scalar fluxes. can be non-linear in We want the model for b l ki to satisfy: Continuity: b k ki = 0, Normalization: bl kk = u lθ. STRESS TRANSPORT MODELLING 2 p.19

A Linear φ iθ Model A linear form for b l ki could be written b l ki = α 1 u l θδ ik + α 2 ( uk θδ li + u i θδ lk ) Application of the above constraints leads to the linear QI model (Launder, 1973): φ iθ2 = 0.8u k θ U i x k 0.2u k θ U k x i But this does not satisfy the 2-component limit. STRESS TRANSPORT MODELLING 2 p.20

Non-Linear φ iθ2 Model 1 Including all cubic terms in b l ki leads to: b l ki = α 1 u l θδ ik + α 2 ( uk θδ li + u i θδ lk ) + α3 u l θa ik + α 4 ( uk θa li + u i θa lk ) + + α 7 u m θa ml a ik + α 8 u m θ (a mk a il + a mi a kl ) + + u n θa mn (α 13 a ml δ ik + α 14 (a mk δ li + a mi δ lk )) To apply the 2-component limit: Shih & Lumley (1985) employed the Schwarz inequality constraint: ( uα θ ) 2 u 2 α θ 2. This leads to both free coefficients in φ ij being determined as zero. Instead, at UMIST, have ensured that nett contribution P 2θ + φ 2θ = 0 when u 2 = 0. This results in the condition U k x l b l k2 = 1 2 u lθ U 2 x l when u 2 = 0 STRESS TRANSPORT MODELLING 2 p.21

Non-Linear φ iθ2 Model 2 Application of continuity, normalization, and 2-component limit results in all coefficients in b l ki being determined: φ iθ2 = 0.8u k θ U i 0.2u k θ U k + (1/3) ɛ x k x i k u iθ P ɛ ) 0.4u k θa il ( Uk x l + U l x k + 0.1u k θa ik a ml ( Um x l + U l x m 0.1u k θ (a im P mk + 2a mk P im ) /k ( Uk + 0.15a ml + U ) l (amku i θ a miu k θ ) x l x k 0.05a ml [ 7a mk ( u i θ U k x l ) + u k θ U i x l ) u k θ ( )] U i U i a ml + a mk x k x l STRESS TRANSPORT MODELLING 2 p.22

Non-Linear φ iθ1 Model A non-linear extension of Monin s (1965) return to isotropy for φ iθ1 : ε φ iθ1 = c θ1 k u iθ c ε θ1 k a iju j θ c ε θ1 k a ika kj u j θ c ε θ1 k A 2u i θ This satisfies realizability for any coefficient values. We found it beneficial to include some mean scalar gradient term for shear flows with strong scalar gradients: φ iθ1 = c θ1 ε k c θ1t Rka ij T x j [ ui θ(1 + c θ1a 2 ) + c θ1a ik u k θ + c θ1a ik a kj u j θ ] where c θ1 = 1.7R 1/2 [ 1 + 1.2(A 2 A) 1/2] c θ1 = 0.8 c θ1 = 1.1 c θ1 = 0.6 c θ1t = 0.2A 1/2 STRESS TRANSPORT MODELLING 2 p.23

Homogeneous Flows 1 Stress anisotropy development in simple shear flow: Note that the basic RSM (from the previous lecture) would return u 2 2 = u2 3. STRESS TRANSPORT MODELLING 2 p.24

Scalar fluxes in simple shear flow: Weakly Strained Homogeneous Flows 2 Strongly Strained - - - : Basic RSM : TCL STRESS TRANSPORT MODELLING 2 p.25

Free Jets Experiments show that an axisymmetric free jet spreads more slowly than a plane one. Standard models (k-ε and the basic RSM) predict the opposite. Dynamic Field Thermal Field Plane Jet Round Jet Plane Jet Round Jet Experiments 0.110 0.095 0.140 0.110 Basic RSM 0.107 0.112 0.145 0.131 TCL Model 0.110 0.102 0.132 0.127 The TCL model does give the correct relative spreading rates. STRESS TRANSPORT MODELLING 2 p.26

Summary In this lecture we have seen how the 2-component limit can be built into second-moment models by ensuring correct behaviour of transport equation when one stress vanishes. Anisotropy invariants can be used to enforce the limit and to make model respond to different types of turbulence. Examples of the development of TCL models for stresses and scalar fluxes have been outlined. It has been shown that TCL models can bring predictive improvements, even in free flows. Further applications will be shown in seminars next week. STRESS TRANSPORT MODELLING 2 p.27

References Fu, S., 1988 Computational Modelling of Turbulent Swirling Flows with Second-Moment Closures, PhD Thesis, Faculty of Technology, University of Manchester. Kim, J., Moin, P., Moser, R., 1987 Turbulence statistics in fully developed channel flow at low Reynolds number, J. Fluid Mech., vol 177, pp 133-166. Launder, B.E., 1973 Scalar property transport by turbulence, Report HTS/73/26, Mech. Eng. Dept., Imperial College, London. Launder, B.E., Reece, G.J., Rodi, W., 1975 Progress in the development of a Reynolds stress turbulence closure, J. Fluid Mech., vol 68, p 537. Lumley, J., 1978 Computational modelling of turbulent flows, Adv. Appl. Mech., vol 18, pp 123-176. Monin, A.S., 1965 On the symmetry of turbulence in the surface layer of air, Izv. Atm. Oceanic Phys., vol 1, p 45. Rotta, J., 1951 Statistische Theorie nichthomogener Turbulenz, Zeitschrift für Physik, vol 129, pp 547-572. Schumann, U., 1977, Realizability of Reynolds stress turbulence models, Phys. Fluids, vol 20, pp 721-725. Shih, T-S., Lumley, J.L., 1985 Modeling of pressure correlation terms in Reynolds stress and scalar flux equations, Report FD-85-03, Sibley School of Mechanical and Aerospace Eng., Cornell University. STRESS TRANSPORT MODELLING 2 p.28