Penalty Methods for Numerical Approximations of Optimal Boundary Flow Control Problems

Abstract

This article is concerned with penalty methods for solving optimal Dirichlet control problems governed by the steady-state and time-dependent Navier-Stokes equations. We present, in two different versions, the penalized methods for solving the steady-slate Dirichlet control problems. These approaches are implemented and compared numerically. We also generalize the penalty methods to the time-dependent case. Scmidiscrete and fully discrete approximations of time-dependent Dirichlet control problems are discussed and implemented. Numerical results for solving both the steady-state and the time dependent Dirichlet control problems are reported.     

Computations of Optimal Controls for Incompressible Flows

Abstract

This paper is concerned with numerical solutions of optimal control problems for unsteady, viscous, incompressible flows. In general, controls can be of the distributed type (external body force) or Dirichlet type 7lpar;e.g., boundary velocity). Here, wc only consider the former case, although most of what we present is also applicable to the latter. Two different optimization objectives and associated solution methodologies are described. One involves a global-in-time functional, the other a local-in-time functional. Which method is preferred depends on the specific application. Some test computational results are presented.     

Optimal Control in Navier-Stokes Equations

This paper presents a formulation for optimal control of a forced convection flow. The state equation that governs the forced convection flow can be expressed as the incompressible Navier-Stokes equations and energy equation. The optimal control can be formulated as finding a control force to minimize a performance function that is defined to evaluate a control object. The stabilized finite element method is used for the spatial discretization, while the Crank-Nicolson scheme is used for the temporal discretization. The Sakawa-Shindo method, which is an iterative procedure, is applied for minimizing the performance function.     

Reduced Basis Method for Optimal Control of Unsteady Viscous Flows

In this article we discuss the reduced basis method (RBM) for optimal control of unsteady viscous flows. RBM is a reduction method in which one can achieve the versatility of the finite element method or another for that matter and gain significant reduction in the number of degrees of freedom. The essential idea in this method is to define a reduced order subspace spanned by few basis elements and then obtain the solution via a Galerkin projection. We present several ways to define this subspace. Feasibility of the approach is demonstrated on two boundary control problems in cavity and wall bounded channel flows. Control action is effected through boundary surface movement on part of the solid wall. Application of RBM to the control problems leads to finite dimensional optimal control problems which are solved using Newton's method. Through computational experiments we demonstrate the feasibility and applicability of the reduced basis method for control of unsteady viscous flows.     

Shape Optimization of Body Located in Incompressible Viscous Flow Based on Optimal Control Theory

This paper presents a numerical method of shape optimization of a body located in an incompressible viscous flow described by the Stokes and Oseen equations. The purpose of this study is to find the optimal shape that minimizes the fluid forces subjected to the body. The formulation of the shape optimization is based on the optimal control theory. The first thing that should be carried out in the optimal control theory is to define a performance function, which expresses the optimal shape. In this study, the fluid forces minimization problem is treated, i.e. fluid forces are directly used in the performance function. The performance function must be minimized subject to the basic equation. The optimal shape, which minimizes the fluid force, is pursued in this paper. This problem can be transformed into the minimization problem without constraint conditions by the Lagrange multiplier. As a numerical example, drag force minimization problems of a body located in low Reynolds number flows are carried out.     

Precessing Vortex Structures in Turbulent Flow within Rotor-Stator Disc Cavities

Several studies of enclosed turbulent flows within rotating discs or cylinders (e.g. [6, 7]) have revealed that, while the geometry may be strictly axisymmetric, it is possible for non-axisymmetric flow patterns to be created within the space. Here we report a visualization study off low induced in the cavity formed between two discs, one rotating, the other stationary. This is an idealization of the flow configuration that occurs between successive stages in the `hot section' of a gas turbine. Such rotor-stator cavities have hitherto been regarded as creating asymmetric flow pattern but Owen [8] has conjectured that the failure to predict heat transfer coefficients accurately for certain radius-to-height ratios may indicate that here, too, organized rotating vortex structures were playing a crucial role. The present study has made an experimental visualization of this flow over a range of conditions in order to test this conjecture and to help guide future numerical explorations. The apparatus comprised a rotating disc over which is fitted a Perspex stationary disc and shroud. The lower disc was rotated for a number of distinct speeds between 30 and 120 rpm and for two ratios of gap-height to radius (H/R). The spin Reynolds number based on gap height and maximum rotational speed, ρΩRh/μ, ranged from 3.7 × 10E4 to 2.24 × 10E5. The flow structures were visualized by injecting ink through a small hypodermic tube at various radii and depths within the cavity and recording the ensuing dye streaks with a video camera mounted above the discs. The results show that, for a wide range of conditions,structured flow with large-scale vortices does indeed arise, the number of vortices diminishing as the spin Reynolds number is increased. The two-vortex S-shaped pattern is stable over a wide range of conditions but three, five and seven vortices have also been observed. These results suggest that an accurate numerical simulation of the flow within rotor-stator disc cavities may require unsteady,three-dimensional CFD modelling over at least certain ranges of flow parameters. This revised version was published online in July 2006 with corrections to the Cover Date.     

Adjoint Equation-Based Methods for Control Problems in Incompressible, Viscous Flows

A review of adjoint equation-based methodologies for viscous,incompressible flow control and optimization problems is given and illustrated by a drag minimization example. A number of approaches to ameliorating the high storage and CPU costs associated with straightforward implementations of adjoint equation based methodologies are discussed. Other issues, including the relative merits of the differentiate-then-discretize and discretize-then-differentiate approaches to deriving discrete adjoint equations, the incorporation of side constraints into adjoint equation-based methodologies, and inaccuracies that occur due to differentiations at the boundary, are also discussed. This revised version was published online in July 2006 with corrections to the Cover Date.     

Velocity Measurements of a Free-Surface Turbulent Flow Penetrating a Porous Medium Composed of Uniform-Size Spheres

We present a laboratory experimental investigation of the interaction between the turbulent flow in an open channel and the turbulent flow within its very permeable bed. The bed was composed of uniform-size spheres packed in a cubic pattern. Fluid velocities were measured by Particle Image Velocimetry (PIV), which allowed us to investigate the spatial distribution of the time-averaged flow properties in the zone where they are strongly affected by the geometry of the rough bed. We investigate the effect of bed porosity on these flow properties by comparing the results of two experimental configurations: one with an impermeable bed composed of a single layer of spheres and another with a permeable bed composed of five layers. For the latter case, PIV measurements of velocities were also carried out inside two pores adjacent to the bed surface. This data provides an insight into the mechanisms of momentum transfer between the turbulent open channel flow and the turbulent flow within its very permeable bed.     

Controlling the Turbulent Flow Past a Thick Airfoil by Means of Flow Enhancement in Vortex Cells Using Suction from Central Body Surfaces

On the basis of a numerical simulation of the turbulent steady-state flow past a thick airfoil with vortex cells built into the body contour, an unconventional technique for controlling flow separation by means of distributed suction from central bodies embedded in the cells is analyzed over a wide range of Reynolds numbers and suction velocities.     

An implicit pseudo‐spectral multi‐domain method for the simulation of incompressible flows

A pseudo‐spectral method for the solution of incompressible flow problems based on an iterative solver involving an implicit treatment of linearized convective terms is presented. The method allows the treatment of moderately complex geometries by means of a multi‐domain approach and it is able to cope with non‐constant fluid properties and non‐orthogonal problem domains. In addition, the fully implicit scheme yields improved stability properties as opposed to semi‐implicit schemes commonly employed. Key components of the method are a Chebyshev collocation discretization, a special pressure–correction scheme, and a restarted GMRES method with a preconditioner derived from a fast direct solver. The performance of the proposed method is investigated by considering several numerical examples of different complexity, and also includes comparisons to alternative solution approaches based on finite‐volume discretizations. Copyright © 2003 John Wiley & Sons, Ltd.     

二维曲壁扩压器内湍流边界层分离的研究

本文应用二维双色四光束激光多普勒测速仪和压强探针样细地测量了二维非对称曲面扩张通道内的不可压湍流边界层分离流动，得到了时均速度和雷诺剪应力以及正反向间歇流动因子和静压分布。实验结果分析表明；湍流边界层分离时，沿边界层高度方向存在着明显的压强差。压强差的极小值对应于位移厚度曲率的极大值和瞬时间歇分离点。Ｂａｒｄｉｎａ对数尾迹律可以较好地描述瞬时间歇分离点之前的边界层速度分布，但无法描述分离的边界层速     

不可压缩机翼绕流的有限谱法计算

结合有限谱QUICK格式求解不可压缩粘性流问题。这一格式用于模拟不同攻角下的NACA1200机翼绕流问题。利用体积力,提出了将流场速度从0加速到来流速度的方法。区别于传统的压力梯度为零的边界条件,推导出一个更精确的压力边界条件。为使速度散度保持为零,在泊松方程中给速度散度一个特殊的处理。这一成果说明了有限谱法不但具有很高的精度,而且能灵活地和其他格式一起构造出新的格式,从而成功地应用到复杂流场不可压缩流动的数值计算中。     

Effect of Interphase Heat Transfer on the Stability of a Turbulent Multicomponent Flow in a Vortex

The parameters of an axisymmetric turbulent two-phase swirling flow of a viscous heat-conducting gas containing a liquid dispersed phase in the presence of water vapor condensation on the particles are calculated. For the dispersed phase, a model taking into account the variation of the vapor concentration and the particle size due to condensation or evaporation is proposed. The distributions of the parameters of the basic unperturbed flow obtained numerically are used in the numerical solution of the linear problem of hydrodynamic stability within the time-dependent formulation. The parameters of small-amplitude harmonic perturbations propagating along the vortex axis are investigated in the linear formulation. A significant effect of heat release in the gas due to water vapor condensation on the parameters of the neutral perturbations and the neutral-stability curves is detected.     

Numerical Approaches to Optimal Control of a Model Equation for Shear Flow Instabilities

We investigate two different discretization approaches of a model optimal-control problem, chosen to be relevant for control of instabilities in shear flows. In the first method, a fully discrete approach has been used, together with a finite-element spatial discretization, to obtain the objective function gradient in terms of a discretely-derived adjoint equation. In the second method, Chebyshev collocation is used for spatial discretization, and the gradient is approximated by discretizing the continuously-derived adjoint equation. The discrete approach always results in a faster convergence of the conjugate-gradient optimization algorithm. Due to the shear in the convective velocity, a low diffusivity in the problem complicates the structure of the computed optimal control, resulting in particularly noticeable differences in convergence rate between the methods. When the diffusivity is higher, the control becomes less complicated, and the difference in convergence rate reduces. The use of approximate gradients results in a higher sensitivity to the degrees of freedom in time. When the system contains a strong instability, it only takes a few iteration to obtain an effective control for both methods,even if there are differences in the formal convergence rate. This indicates that it is possible to use the approximative gradients of the objective function in cases where the control problem mainly consists of controlling strong instabilities. This revised version was published online in July 2006 with corrections to the Cover Date.     

Viscous flows in corner regions: Singularities and hidden eigensolutions

Numerical issues arising in computations of viscous flows in corners formed by a liquid–fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the ‘moving contact‐line problem’ and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite‐element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh‐dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh‐independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite‐element method in the context of the present problem are discussed. Copyright © 2009 John Wiley & Sons, Ltd.     

Turbulent Channel Flow with an Artificial Two-Dimensional Wall Layer

Turbulent flow of an incompressible fluid in a plane channel with parallel walls is considered. The three-dimensional time-dependent Navier-Stokes equations are solved numerically using the spectral finite-difference method. An artificial force which completely suppresses lateral oscillations of the velocity is introduced in the near-wall zone (10 % of the channel half-width in the neighborhood of each wall). Thus, the three-dimensional flow zone, in which turbulent oscillations can develop, is separated from the wall by a fluid layer. It is found that the elimination of three-dimensionality in the neighborhood of the walls leads to a significant reduction in the drag. However, complete laminarization does not occur. The flow in the stream core remains turbulent and can be interpreted as a turbulent flow in a channel with walls located on the boundary of the two-dimensional layer and traveling at the local mean-flow velocity. The oscillations developing inside the two-dimensional layer, which have significant amplitude, distort the flow only in the adjacent zone. Beyond this zone the distributions of the mean characteristics and the structure of instantaneous fields completely correspond to ordinary turbulent flow in a channel with rigid walls. The results obtained confirm the hypothesis of the unimportance of the no-slip boundary conditions for the fluctuating velocity component in the mechanism of onset and self-maintenance of turbulence in wall flows.     

Numerical Study of a Turbulent Flow Compressed by a Weak Shock Wave

Abstract

Modification of a two-dimensional isotropic trbulent flow by passage of a weak shock wave was numerically studied by the explicit spectral collocation method with the fourier series in the ranges of M₁ = 0.14?0.21 and M_s = 1.06?1.14. The density 1 trbulent kinetic energy increased by 10 to 19 % and density fluctuation increased by 14 to 50 % in proportion to the shock wave mach number. The amplification of turbulent kinetic energy did not change by increase of the turbulent Mach number, while that of density fluctuation decreased from1.35 to 1.15%. The argumentation of turbulent kinetic energy appeared at moderate and high wave numbers of energy spectra and resulted in a reduction of the integral scale by3.33 to 5.4%. Modifications mechanisms of turbulent kinetic energy, density fluctuation and vorticity were verified by the transport equations It was shown that the decreased amplification of the density fluctuation In the increased trbulent Mach number attributed to the unchanged production terms and the negative dilatation correlation behind the shock wave.the vorticity-dilatation term was responsible for more than 80 % of the total vorticity production at the shock front.     

Effect of the Turbulent Energy Gradient and the Presence of a Wall on the Turbulent Process of Momentum Transport

A mathematical model of turbulent transport processes is modified to make allowance for the turbulent energy gradient and the presence of walls. The modification consists in making the variance tensor in the Gaussian probability density distribution for the initial mole velocities anisotropic for nonzero turbulent energy gradient and a ratio of the turbulence scale to the distance from the wall of the order of unity. Formulas for the variance tensor components are derived and the empirical coefficients of these formulas are determined. The expression for the dimensionless turbulent friction stress is compared with experimental data for three boundary-layer-type flows, namely, in the wake of a cylinder, in the boundary layer on a flat plate, and in a channel with parallel walls.     

A segregated implicit solution algorithm for 2D and 3D laminar incompressible flows

A segregated algorithm for the solution of laminar incompressible, two- and three-dimensional flow problems is presented. This algorithm employs the successive solution of the momentum and continuity equations by means of a decoupled implicit solution method. The inversion of the coefficient matrix which is common for all momentum equations is carried out through an approximate factorization in upper and lower triangular matrices. The divergence-free velocity constraint is satisfied by formulating and solving a pressure correction equation. For the latter a combined application of a preconditioning technique and a Krylov subspace method is employed and proved more effecient than the approximate factorization method. The method exhibits a monotonic convergence, it is not costly in CPU time per iteration and provides accurate solutions which are independent of the underrelaxation parameter used in the momentum equations. Results are presented in two- and three-dimensional flow problems.