Continuous adjoint approach to the Spalart-Allmaras turbulence model for incompressible flows

A continuous adjoint formulation for the computation of the sensitivities of integral functions used in steady-flow, incompressible aerodynamics is presented. Unlike earlier continuous adjoint methods, this paper computes the adjoint to both the mean-flow and turbulence equations by overcoming the frequently made assumption that the variation in turbulent viscosity can be neglected. The development is based on the Spalart-Allmaras turbulence model, using the adjoint to the corresponding differential equation and boundary conditions. The proposed formulation is general and can be used with any other integral function. Here, the continuous adjoint method yielding the sensitivities of the total pressure loss functional for duct flows with respect to the normal displacements of the solid wall nodes is presented. Using three duct flow problems, it is demonstrated that the adjoint to the turbulence equations should be taken into account to compute the sensitivity derivatives of this functional with high accuracy. The so-computed derivatives almost coincide with “reference” sensitivities resulting from the computationally expensive direct differentiation. This is not, however, the case of the sensitivities computed without solving the turbulence adjoint equation, which deviates from the reference values. The role of all newly appearing terms in the adjoint equations, their boundary conditions and the gradient expression is investigated, significant and insignificant terms are identified and a study on the Reynolds number effect is included.     

An implicit, exact dual adjoint solution method for turbulent flows on unstructured grids

An implicit algorithm for solving the discrete adjoint system based on an unstructured-grid discretization of the Navier-Stokes equations is presented. The method is constructed such that an adjoint solution exactly dual to a direct differentiation approach is recovered at each time step, yielding a convergence rate which is asymptotically equivalent to that of the primal system. The new approach is implemented within a three-dimensional unstructured-grid framework and results are presented for inviscid, laminar, and turbulent flows. Improvements to the baseline solution algorithm, such as line-implicit relaxation and a tight coupling of the turbulence model, are also presented. By storing nearest-neighbor terms in the residual computation, the dual scheme is computationally efficient, while requiring twice the memory of the flow solution. The current implementation allows for multiple right-hand side vectors, enabling simultaneous adjoint solutions for several cost functions or constraints with minimal additional storage requirements, while reducing the solution time compared to serial applications of the adjoint solver. The scheme is expected to have a broad impact on computational problems related to design optimization as well as error estimation and grid adaptation efforts.     

Multi-point aerodynamic shape optimization of cars based on continuous adjoint

This article presents a continuous adjoint-enabled, gradient-based optimization tool for multi-point, multi-objective industrial optimization problems and its application to the shape optimization of a concept car. Apart from the adjoint to the incompressible Reynolds-averaged Navier–Stokes equations, the adjoint to the Spalart–Allmaras turbulence model equation is also solved, in order to support the optimization with accurate gradients. Part of the mathematical development related to the sensitivity derivative terms resulting from the differentiation of the Reynolds-averaged Navier–Stokes (RANS) variant of the Spalart–Allmaras model when using an adjoint formulation consisting of field integrals is presented for the first time in the literature. In the industrial application, two operating points are considered, corresponding to two flow velocity angles with respect to the car symmetry plane, with a different objective (drag and yaw moment coefficients) for each of them. With the aforesaid targets, the Pareto front of optimal solutions is computed and discussed. Each point on this front is computed by minimizing a single objective function, resulting from the linear combination of the objective functions defined on the different operating points, using appropriate weights. Finally, some of the Pareto front members are re-evaluated using delayed detached eddy simulation (DDEs). The overall optimization tool is developed in the open-source CFD toolbox OpenFOAM.

     

Aerodynamic Shape Optimization Using First and Second Order Adjoint and Direct Approaches

This paper focuses on discrete and continuous adjoint approaches and direct differentiation methods that can efficiently be used in aerodynamic shape optimization problems. The advantage of the adjoint approach is the computation of the gradient of the objective function at cost which does not depend upon the number of design variables. An extra advantage of the formulation presented below, for the computation of either first or second order sensitivities, is that the resulting sensitivity expressions are free of field integrals even if the objective function is a field integral. This is demonstrated using three possible objective functions for use in internal aerodynamic problems; the first objective is for inverse design problems where a target pressure distribution along the solid walls must be reproduced; the other two quantify viscous losses in duct or cascade flows, cast as either the reduction in total pressure between the inlet and outlet or the field integral of entropy generation. From the mathematical point of view, the three functions are defined over different parts of the domain or its boundaries, and this strongly affects the adjoint formulation. In the second part of this paper, the same discrete and continuous adjoint formulations are combined with direct differentiation methods to compute the Hessian matrix of the objective function. Although the direct differentiation for the computation of the gradient is time consuming, it may support the adjoint method to calculate the exact Hessian matrix components with the minimum CPU cost. Since, however, the CPU cost is proportional to the number of design variables, a well performing optimization scheme, based on the exactly computed Hessian during the starting cycle and a quasi Newton (BFGS) scheme during the next cycles, is proposed.     

On Visualizing Continuous Turbulence Scales

Turbulent flows are multi‐scale with vortices spanning a wide range of scales continuously. Due to such complexities, turbulence scales are particularly difficult to analyse and visualize. In this work, we present a novel and efficient optimization‐based method for continuous‐scale turbulence structure visualization with scale decomposition directly in the Kolmogorov energy spectrum. To achieve this, we first derive a new analytical objective function based on integration approximation. Using this new formulation, we can significantly improve the efficiency of the underlying optimization process and obtain the desired filter in the Kolmogorov energy spectrum for scale decomposition. More importantly, such a decomposition allows a ‘continuous‐scale visualization’ that enables us to efficiently explore the decomposed turbulence scales and further analyse the turbulence structures in a continuous manner. With our approach, we can present scale visualizations of direct numerical simulation data sets continuously over the scale domain for both isotropic and boundary layer turbulent flows. Compared with previous works on multi‐scale turbulence analysis and visualization, our method is highly flexible and efficient in generating scale decomposition and visualization results. The application of the proposed technique to both isotropic and boundary layer turbulence data sets verifies the capability of our technique to produce desirable scale visualization results.     

Adjoint network method applied to the performance sensitivities of microwave amplifiers

This work focuses on the performance sensitivities of microwave amplifiers using the “adjoint network and adjoint variable” method, via “wave” approaches, which includes sensitivities of the transducer power gain, noise figure, and magnitudes and phases of the input and output reflection coefficients. The method can be extended to sensitivities of the other performance measure functions. The adjoint‐variable methods for design‐sensitivity analysis offer computational speed and accuracy. They can be used for efficiency‐based gradient optimization, in tolerance and yield analyses. In this work, an arbitrarily configured microwave amplifier is considered: firstly, each element in the network is modeled by the scattering matrix formulation, then the topology of the network is taken into account using the connection scattering‐matrix formulation. The wave approach is utilized in the evaluation of all the performance‐measurement functions, then sensitivity invariants are formulated using Tellegen's theorem. Performance sensitivities of the T‐ and Π‐types of distributed‐parameter amplifiers are considered as a worked example. The numerical results of T‐ and Π‐type amplifiers for the design targets of noise figure F_req = 0.46 dB ? 1,12 and V_ireq = 1, G_Treq = 12 dB ? 15.86 in the frequency range 2–11 GHz are given in comparison to each other. Furthermore, analytical methods of the “gain factorisation” and “chain sensitivity parameter” are applied to the gain and noise sensitivities as well. In addition, “numerical perturbation” is applied to calculation of all the sensitivities. © 2006 Wiley Periodicals, Inc. Int J RF and Microwave CAE, 2006.     

An Optimized Low-Dissipation Monotonicity-Preserving Scheme for Numerical Simulations of High-Speed Turbulent Flows

This paper presents an optimized low-dissipation monotonicity-preserving (MP-LD) scheme for numerical simulations of high-speed turbulent flows with shock waves. By using the bandwidth dissipation optimization method (BDOM), the linear dissipation of the original MP scheme of Suresh and Huynh (J. Comput. Phys. 136, 83–99, 1997) is significantly reduced in the newly developed MP-LD scheme. Meanwhile, to reduce the nonlinear dissipation and errors, the shock sensor of Ducros et al. (J. Comput. Phys. 152, 517–549, 1999) is adopted to avoid the activation of the MP limiter in regions away from shock waves. Simulations of turbulent flows with and without shock waves indicate that, in comparison with the original MP scheme, the MP-LD scheme has the same capability in capturing shock waves but a better performance in resolving small-scale turbulence fluctuations without introducing excessive numerical dissipation, which implies the MP-LD scheme is a valuable tool for the direct numerical simulation and large eddy simulation of high-speed turbulent flows with shock waves.     

Density based topology optimization of turbulent flow heat transfer systems

The focus of this article is on topology optimization of heat sinks with turbulent forced convection. The goal is to demonstrate the extendibility, and the scalability of a previously developed fluid solver to coupled multi-physics and large 3D problems. The gradients of the objective and the constraints are obtained with the help of automatic differentiation applied on the discrete system without any simplifying assumptions. Thus, as demonstrated in earlier works of the authors, the sensitivities are exact to machine precision. The framework is applied to the optimization of 2D and 3D problems. Comparison between the simplified 2D setup and the full 3D optimized results is provided. A comparative study is also provided between designs optimized for laminar and turbulent flows. The comparisons highlight the importance and the benefits of full 3D optimization and including turbulence modeling in the optimization process, while also demonstrating extension of the methodology to include coupling of heat transfer with turbulent flows.     

Direct numerical simulation of transitional flow at high Mach number coupled with a thermal wall model

In transitional and turbulent high speed boundary-layer flows the wall thermal boundary conditions play an important role and in many cases an assumption of a constant temperature or a specified heat flux may not be appropriate for numerical simulations. In this paper we extend a formulation for direct numerical simulation of compressible flows to include a thin plate that is thermally fully coupled to the flow. Even without such thermal coupling compressible flows with shock waves and turbulence represent a challenge for numerical methods. In this paper we review the scaling properties of algorithms, based on explicit high-order finite differencing combined with shock capturing, that are suitable for dealing with such flows. An application is then considered in which an isolated roughness element is of sufficient height to trigger transition in the presence of acoustic forcing. With the thermal wall model included it is observed that the plate heats up sufficiently during the simulation for the transition process to be halted and the flow consequently re-laminarises.     

Numerical simulation of three-dimensional turbulent separated and reattaching flows using a modified turbulence model

A modified version of k-ε model is proposed through modification of the damping function of eddy viscosity that incorporates the effect of wall proximity in the near the wall region and the effect of non-equilibrium away from the wall together with the simple model functions in the ε equation. The proposed turbulence model is validated with the available experimental data of reattachment length, mean streamwise velocity distribution, turbulence intensity profile, and wall static pressure coefficient in the turbulent backward-facing step flows. The predicted results with the present model are in good agreement with the experiments. Computed results reveal that the reattachment length (recirculation zone) and the wall static pressure are decreased with increasing inlet velocity. And the asymmetric distributions of the reattachment point, cross-section view of velocity vector, streamwise skin friction coefficient, and turbulent kinetic energy demonstrate the important three-dimensional side-wall effect in an insufficient aspect ratio channel flow.     

Hybrid spectral-element-low-order methods for incompressible flows

In this article we present a new formulation for coupling spectral element discretizations to finite difference and finite element discretizations addressing flow problems in very complicated geometries. A general iterative relaxation procedure (Zanolli patching) is employed that enforcesC ¹ continuity along the patching interface between the two differently discretized subdomains. In fluid flow simulations of transitional and turbulent flows the high-order discretization (spectral element) is used in the outer part of the domain where the Reynolds number is effectively very high. Near rough wall boundaries (where the flow is effectively very viscous) the use of low-order discretizations provides sufficient accuracy and allows for efficient treatment of the complex geometry. An analysis of the patching procedure is presented for elliptic problems, and extensions to incompressible Navier-Stokes equations are implemented using an efficient high-order splitting scheme. Several examples are given for elliptic and flow model problems and performance is measured on both serial and parallel processors.     

Synthetic Controllable Turbulence Using Robust Second Vorticity Confinement

Capturing fine details of turbulence on a coarse grid is one of the main tasks in real‐time fluid simulation. Existing methods for doing this have various limitations. In this paper, we propose a new turbulence method that uses a refined second vorticity confinement method, referred to as robust second vorticity confinement, and a synthesis scheme to create highly turbulent effects from coarse grid. The new technique is sufficiently stable to efficiently produce highly turbulent flows, while allowing intuitive control of vortical structures. Second vorticity confinement captures and defines the vortical features of turbulence on a coarse grid. However, due to the stability problem, it cannot be used to produce highly turbulent flows. In this work, we propose a robust formulation to improve the stability problem by making the positive diffusion term to vary with helicity adaptively. In addition, we also employ our new method to procedurally synthesize the high‐resolution flow fields. As shown in our results, this approach produces stable high‐resolution turbulence very efficiently.     

A Functional Approach to the Visual Simulation of Gaseous Turbulence

This paper presents a functional method for the visual simulation of 2-D or 3-D turbulent gaseous motion by using time-varying fractals. The used function incorporates results from the “spectral theory of turbulence”, thereby providing a physics-based approach adapted to the needs of computer graphics. The involved turbulence function is band-limited, continuous, differentiable, anisotrop, and smooth, provides different fractal dimensions along each axis, may be evaluated locally with different parameters, and requires only minimal storage space, thus supporting an implementation on large parallel processing networks with small nodes. Inhomogeneity in the form of local disturbances of the turbulence field may also be easily considered. The parameters used to describe turbulent motion are rather intuitive, so that they may be utilized easily by users. Examples for modeling different types of clouds and fire are given.     

Applications of regional strain energy in compliant structure design for energy absorption

Topology optimization of regional strain energy is studied in this paper. Unlike the conventional mean compliance formulation, this paper considers two main functions of structure: rigidity and compliance. For normal usages, rigidity is chosen as the design objective. For compliant design, a portion of the structure absorbs energy, while another part maintains the structural integrity. Therefore, we implemented a regional strain energy formulation for topology optimization. Sensitivity to regional strain energy is derived from the adjoint method. Numerical results from the proposed formulation are presented.     

Topology optimization of steady and unsteady incompressible Navier–Stokes flows driven by body forces

This paper presents the topology optimization method for the steady and unsteady incompressible Navier–Stokes flows driven by body forces, which typically include the constant force (e.g. the gravity) and the centrifugal and Coriolis forces. In the topology optimization problem, the artificial friction force with design variable interpolated porosity is added into the Navier–Stokes equations as the conventional method, and the physical body forces in the Navier–Stokes equations are penalized using the power-law approach. The topology optimization problem is analyzed by the continuous adjoint method, and solved by the finite element method in conjunction with the gradient based approach. In the numerical examples, the topology optimization of the fluidic channel, mass distribution of the flow and local velocity control are presented for the flows driven by body forces. The numerical results demonstrate that the presented method achieves the topology optimization of the flows driven by body forces robustly.     

Rapid design closure of microwave components by means of feature‐based optimization and adjoint sensitivities

In this article, fast design closure of microwave components using feature‐based optimization (FBO) and adjoint sensitivities is discussed. FBO is one of the most recent optimization techniques that exploits a particular structure of the system response to “flatten” the functional landscape handled during the optimization process, which leads to reducing its computational complexity. When combined with gradient‐based search involving adjoint sensitivities, the design cost becomes even lower, allowing us to find the optimum design using just a few electromagnetic (EM) simulations of the structure at hand. Here, operation and performance of the algorithm is demonstrated using a waveguide filter and a miniaturized microstrip rat‐race coupler (RRC). Comparative studies indicate considerable savings that can be achieved even compared with adjoint‐based gradient search. In case of RRC, numerical results are supported by experimental validation.     

Optimum material design of minimum structural compliance under seepage constraint

In filtration and chemical engineering industry the load carrying capacity and seepage performances are very important for a successful filter design. We study a two-scale structural design optimization problem to minimize structural compliance under given seepage flow rate and material porosity constraints. Structural size, shape and topology are given because of other functional requirements. Structural material used is macro homogeneous porous material with periodic microstructure and is to be designed. Since structural compliance and seepage performances in macro-scale are implicit functions of material microstructural topology, it becomes a two-scale design optimization problem. The cross scale sensitivities are derived by the adjoint method. A new volume preserving nonlinear density filter is proposed which makes the process of optimization iteration more stable. The optimization problem is solved by GCMMA. Examples under the equality constraints of different seepage flow rate are presented to illustrate the effectiveness of two-scale design optimization formulation and solution approach.     

Visualization of vorticity and vortices in wall-bounded turbulent flows

This study was initiated by the scientifically interesting prospect of applying advanced visualization techniques to gain further insight into various spatio-temporal characteristics of turbulent flows. The ability to study complex kinematical and dynamical features of turbulence provides means of extracting the underlying physics of turbulent fluid motion. The objective is to analyze the use of a vorticity field line approach to study numerically generated incompressible turbulent flows. In order to study the vorticity field, we present a field line animation technique which uses a specialized particle advection and seeding strategy. Efficient analysis is achieved by decoupling the rendering stage from the preceding stages of the visualization method. This allows interactive exploration of multiple fields simultaneously, which sets the stage for a more complete analysis of the flow field. Multifield visualizations are obtained using a flexible volume rendering framework which is presented in this paper. Vorticity field lines have been employed as indicators to provide a means to identify "ejection" and "sweep" regions; two particularly important spatio-temporal events in wall-bounded turbulent flows. Their relation to the rate of turbulent kinetic energy production and viscous dissipation, respectively, have been identified.     

Adjoint sensitivity analysis and optimization of hysteretic dynamic systems with nonlinear viscous dampers

In this paper we discuss the adjoint sensitivity analysis and optimization of hysteretic systems equipped with nonlinear viscous dampers and subjected to transient excitation. The viscous dampers are modeled via the Maxwell model, considering at the same time the stiffening and the damping contribution of the dampers. The time-history analysis adopted for the evaluation of the response of the systems relies on the Newmark-β time integration scheme. In particular, the dynamic equilibrium in each time-step is achieved by means of the Newton-Raphson and the Runge-Kutta methods. The sensitivity of the system response is calculated with the adjoint variable method. In particular, the discretize-then-differentiate approach is adopted for calculating consistently the sensitivity of the system. The importance and the generality of the sensitivity analysis discussed herein is demonstrated in two numerical applications: the retrofitting of a structure subject to seismic excitation, and the design of a quarter-car suspension system. The MATLAB code for the sensitivity analysis considered in the first application is provided as “Supplementary Material”.     

Minimum‐variance control of astronomical adaptive optic systems with actuator dynamics under synchronous and asynchronous sampling

Adaptive optic (AO) systems are now routinely used in ground‐based telescopes to counter the effects of atmospheric turbulence. A deformable mirror (DM) generates a correction wavefront, which is subtracted from the turbulent wavefront using measurements of the residual phase provided by a wavefront sensor (WFS). Minimizing the variance of the residual phase defines a sampled data control problem combining a continuous time minimum‐variance (MV) performance criterion with a discrete‐time controller. For a fairly general class of linear time‐invariant DM and turbulence WFS models, this control problem can be transformed into an equivalent discrete‐time LQ optimization problem involving a set of (discrete‐time) control‐sufficient statistics of the incoming continuous‐time turbulence. This paper shows how to constructively solve this MV problem in the presence of DM's dynamics, starting from continuous‐time models of DM and turbulence. This result is extended to the case of asynchronous DM/WFS sampling. An illustrative application to optimal control of tip‐tilt turbulent modes for the European extremely large telescope in the presence of first‐order DM's dynamics is presented. Copyright © 2010 John Wiley & Sons, Ltd.