Reynolds-Averaged Navier-Stokes (RANS) models are not very accurate for high Reynolds number, compressible jet-in-crossflow interactions. The inaccuracy arises from the use of inappropriate model parameters and model-form error in the RANS model. In this work we pursue the hypothesis that RANS predictions could be significantly improved by using parameters inferred from experimental measurements of a supersonic jet interacting with a transonic crossflow. We formulate a Bayesian inverse problem to estimate 3 RANS parameters (C µ , C ǫ2 , C ǫ1) and use a Markov chain Monte Carlo (MCMC) method to develop a probability density function for them. The cost of MCMC is addressed by developing statistical surrogates for the RANS models. We find that only a subset of the (C µ , C ǫ2 , C ǫ1) space, R, supports realistic flow simulations. We use R as our prior belief when formulating the inverse problem. It is enforced with a classifier in our MCMC solution. We find that the calibrated parameters improve predictions of the entire flowfield substantially, compared to the nominal/literature values of (C µ , C ǫ2 , C ǫ1); further, this improvement is seen to hold for interactions at other Mach numbers and jet strengths for which we have experimental data to provide a comparison. We also quantify the residual error, which is an approximation of the model-form error; it is most easily measured in terms of turbulent stresses.
We propose a Bayesian method to calibrate parameters of a k− RANS model to improve its predictive skill in jet-in-crossflow simulations. The method is based on the hypotheses that (1) informative parameters can be estimated from experiments of flow configurations that display the same, strongly vortical features of jet-in-crossflow interactions and (2) one can construct surrogates of RANS models for judiciously chosen outputs which serve as calibration observables. We estimate three k − parameters, (Cµ, C 2, C 1), from Reynolds stress measurements obtained from an incompressible flow-over-a-square-cylinder experiment. The k − parameters are estimated as a joint probability density function. Jet-incrossflow simulations performed with (Cµ, C 2, C 1) samples drawn from this distribution are seen to provide far better predictions than those obtained with nominal parameter values. We also find a (Cµ, C 2, C 1) combination which provides less than 15% error in a number of performance metrics. In contrast, the errors obtained with nominal parameter values may exceed 60%.
Summary In this paper we present a 3D flux-continuous finite-difference formulation designed for flow simulation of models with nonorthogonal hexahedron grids with general tensor permeability. Our development follows that of Aavatsmark et al.,1 but we do not operate in transformed space. The new 27-point discretization formula has been implemented in a finite-difference reservoir simulator. This stencil has many desirable properties, including collapsing into a consistent form in two dimensions. We demonstrate that there are many practical situations in which neglecting the influence of nonorthogonality and general tensors results in first-order errors in flow predictions. A rigorous implementation for this 27-point difference operator as a control-volume finite-difference method determines the upwinding of convection terms associated with multiphase computations. Results and issues associated with implementation of this operator in a conventional finite-difference reservoir simulator are discussed. As an alternative to directly solving the linear matrix associated with the 27-point stencil of the flux-continuous operator, we examine iterative methods that split the matrix into a 7-point stencil part and a remainder. The 7-point stencil part is solved by a direct or iterative method, with the remainder part updated from the previous timestep or iteration. This split operator may permit retention of the linear solver for the standard 7-point formulation while retaining nonorthogonal grid and tensor information. Introduction Reservoir characterization models continue to grow in geologic and geometric complexity. Highly detailed descriptions of reservoir properties and complex stratigraphic and structural features are being included into real-field applications. This continued expansion in the complexity of reservoir models is stretching the capabilities of conventional, or brick-like, grids. Nonorthogonal grids, on the other hand, can describe geometrically complex surfaces and volumes accurately. While tetrahedron grids are the most flexible grids in three dimensions, their application in reservoir flow simulation is still limited. A number of important prerequisites are necessary in order to benefit from the flexibility of general unstructured grids. These include rigorous discretization schemes, improved solution algorithms, and more powerful computational and visualization resources. Stratigraphic grids (S-grids) offer an alternative, and they are common in flow simulation practice. Based on hexahedra, S-grids can accommodate complex geologic and geometric features while retaining many of the desirable features of structured grids. For example, S-grids aligned with layer-boundary surfaces are generally structured, but nonorthogonal. To translate the benefits of S-grids to more accurate flow predictions in geometrically complex reservoirs, a rigorous treatment is needed. Discretization of the flow equations for domains with nonorthogonal S-grids yields full tensors. Full tensors can also result from considerations related to the scaleup of detailed description of reservoir properties (e.g., permeability). The scaled-up permeability is a full tensor even if the fine-scale description is isotropic2 and the grid is orthogonal. Furthermore, if the simulation grid is not aligned with the principal directions of permeability, the permeability of the simulation grid becomes a full tensor with nonzero off-diagonal terms.3 For some systems, the cross terms of the permeability tensor can be quite important. These include formations containing complex crossbedding, dipping layers not aligned with the coordinate system,2 or extensive fracturing. 4,5 In these cases the simulation model will contain full tensor permeabilities with significant off-diagonal terms, which must be accommodated by the simulator if reservoir flow performance is to be predicted accurately. Edwards and Rogers6 and Aavatsmark et al.7 independently proposed a flux-continuous finite-difference formulation for two-dimensional (2D) flow. Crumpton et al.8 developed a similar formula based on flux continuity at cell boundaries. Lee et al.3 derived an explicit form of the flux-continuous finite-difference method for 2D Cartesian grids with tensor permeability. They also investigated real-field flow problems. Aavatsmark et al.1 recently extended a similar development to three-dimensional (3D) flow. In their development, Aavatsmark et al. translated the problem from real-space to orthogonal Cartesian grid using a generalized coordinate transform. The computations of the discretization scheme are then carried out in transformed space. Following the development of Aavatsmark et al.,1 we derive a 3D, flux-continuous, finite-difference operator designed to handle nonorthogonal grids with heterogeneous full-tensor permeability. Unlike Aavatsmark et al.,1 however, we do not use a coordinate transform. For general 3D hexahedron grids with heterogeneous permeability, the coordinate transformation does not necessarily yield a simpler derivation of the flux-continuous finite-difference formula. Our scheme produces a 27-point stencil for 3D hexahedron grids. The flux continuity conditions among the 27 neighboring cells entails solving a 24×24 matrix. Because the transmissibilities are computed only once in the initialization step of reservoir simulation, the relatively complex transmissibility computation of the flux-continuous method should have a minimal impact on the overall numerical performance.
At 70 miles per hour, overcoming aerodynamic drag represents about 65% of the total energy expenditure for a typical heavy truck vehicle. The goal of this US Department of Energy supported consortium is to establish a clear understanding of the drag producing flow phenomena. This is being accomplished through joint experiments and computations, leading to the 'smart' design of drag reducing devices. This paper will describe our objective and approach, provide an overview of our efforts and accomplishments, and discuss our future direction.
We demonstrate a statistical procedure for learning a high-order eddy viscosity model (EVM) from experimental data and using it to improve the predictive skill of a Reynolds-averaged Navier–Stokes (RANS) simulator. The method is tested in a three-dimensional (3D), transonic jet-in-crossflow (JIC) configuration. The process starts with a cubic eddy viscosity model (CEVM) developed for incompressible flows. It is fitted to limited experimental JIC data using shrinkage regression. The shrinkage process removes all the terms from the model, except an intercept, a linear term, and a quadratic one involving the square of the vorticity. The shrunk eddy viscosity model is implemented in an RANS simulator and calibrated, using vorticity measurements, to infer three parameters. The calibration is Bayesian and is solved using a Markov chain Monte Carlo (MCMC) method. A 3D probability density distribution for the inferred parameters is constructed, thus quantifying the uncertainty in the estimate. The phenomenal cost of using a 3D flow simulator inside an MCMC loop is mitigated by using surrogate models (“curve-fits”). A support vector machine classifier (SVMC) is used to impose our prior belief regarding parameter values, specifically to exclude nonphysical parameter combinations. The calibrated model is compared, in terms of its predictive skill, to simulations using uncalibrated linear and CEVMs. We find that the calibrated model, with one quadratic term, is more accurate than the uncalibrated simulator. The model is also checked at a flow condition at which the model was not calibrated.
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