A stress-strain lag Eddy viscosity model for variable density flow Mahmoud Assad 1 , Robert Prosser 1 , Alistair Revell 1 , and Bertrand Sapa 2 1 Modelling and Simulation Centre, School of MACE, University of Manchester, M13 9PL, UK 2 Electric de France, 6 quai Watier, 78491, Chatou, France. Abstract A modified eddy viscosity model is proposed to codify the misalignment between stress and strain fields for variable density flows. The stress-strain misalignment is quanti- fied by introducing a C as parameter. A transport equa- tion for C as is derived from a full Reynolds stress model (RSM). The C as transport equation is coupled to a stan- dard EVM model (e.g, k - ω SST ). The performance of the proposed model is investigated via a turbulent buoy- ant plume. 1 Introduction Variable density flow has many industrial applications such as in incidental fires and buoyant plumes. Such flows usually include high density fluctuations which sig- nificantly increase the unsteadiness and anisotropy of the flow which, in turn, impacts on flow properties such as Reynolds stresses. Two main types of CFD model may be used to capture flow unsteadiness; Reynolds- averaged Navier-Stokes (RANS), and Large Eddy sim- ulations (LES). Using LES for variable density flow sim- ulation is still computationally expensive for industrial applications due to the fine grids that must be used. RANS models are widely used and are computationally cheaper compared to LES models. RANS models his- torically have been developed for cold flows and then modified for hot flows to include buoyancy effects; many studies have applied Eddy Viscosity Models (especially the k - ε model) to variable density flow [1, 2, 3, 4, 5, 6]. In the derivation of standard eddy viscosity models for steady quasi-equilibrium flows, several fundamental as- sumptions are used, such as isotropy of the normal stresses [7]. These assumptions no longer hold for vari- able density flow due to the anisotropy in Reynolds stress distributions. Reynolds stress models improve Reynolds stress predictions, which leads to improvement in the prediction of turbulence energy distribution [7]. How- ever, Reynolds stress models are computationally ex- pensive due to the requirement for solving an additional five transport equations, and the special numerical treat- ments required to make the calculation converge. An alternative simplified approach has been introduced by Revell [8] to codify the misalignment between induced stress and strain fields and The C as model provides a new EVM which has been applied to cold flow appli- cations, and improved results have been observed when compared to standard EVM approaches. [7, 8, 9, 10]. The C as model is suitable for large mean flow unsteadi- ness, and is cheaper to compute than RSM models [7]. The aims of the present work are; to develop a C as model for variable density flows that predicts the anisotropy and unsteadiness effects and; to quantify the misalignment between the stress and strain tensors for hot flows, by introducing a C as parameter that can be modelled locally in the flow field. This parameter can be introduced to derive a new transport equation based on the Favre-averaged Reynolds stress transport equation. The proposed equation is solved in conjunction with clas- sical EVM models. The model performance is compared to different RANS models for a turbulent buoyant plume. 2 Derivation of stress strain lag model 2.1 Background For classical EVM models the production rate of the tur- bulence kinetic energy P k is related to the square of the strain rate  S ij . In the case of Reynolds stress models, the production rate is related linearly to the strain rate  S ij . Classical EVM models thus overestimate the turbu- lence kinetic energy P k in the presence of high strain rate [9, 11], and different models have been proposed to cor- rect the overestimate, as in the linear production model by Guiment and Laurence [11]. However, such models do not consider the effect of stress-strain lag in the unsteady flow. The C as model introduces the turbulence production rate as P k =¯ ρC as  k  S, where C as is a non-dimensional parameter representing the alignment between the stress and strain fields: C as = -  a ij  S ij   S , (1)  a ij = ] u  i u  j  k - 2 3 δ ij ,  S ij =  ∂  u i ∂x j + ∂  u j ∂x i  , (2) where  a ij is the stress anisotropy,  k = 1 2 ] u  i u  j is the tur- bulent kinetic energy, δ ij is the Kronecker delta function,  S ij is the strain rate and   S =  2  S ij  S ij is the strain invariant. For variable density flow, the strain rate ten- sor is replaced by the traceless form as  S ij =  ∂  u i ∂x j + ∂  u j ∂x i - 1 3 ∂  u k ∂x k δ ij  (3) The effects of stress strain misalignment have been re- viewed in [7, 12]. ERCOFTAC Bulletin 89 1