Optimal flow control for Navier–Stokes equations: drag minimization

Optimal control and shape optimization techniques have an increasing role in Fluid Dynamics problems governed by partial differential equations (PDEs). In this paper, we consider the problem of drag minimization for a body in relative motion in a fluid by controlling the velocity through the body boundary. With this aim, we handle with an optimal control approach applied to the steady incompressible Navier–Stokes equations. We use the Lagrangian functional approach and we consider the Lagrangian multiplier method for the treatment of the Dirichlet boundary conditions, which include the control function itself. Moreover, we express the drag coefficient, which is the functional to be minimized, through the variational form of the Navier–Stokes equations. In this way, we can derive, in a straightforward manner, the adjoint and sensitivity equations associated with the optimal control problem, even in the presence of Dirichlet control functions. The problem is solved numerically by an iterative optimization procedure applied to state and adjoint PDEs which we approximate by the finite element method. Copyright © 2007 John Wiley & Sons, Ltd.     

Shape optimization of a body located in low Reynolds number flow

The purpose of this study is to perform a numerical application of the shape optimization formulation of a body located in an incompressible viscous flow field. The formulation is based on an optimal control theory in which a performance function of the fluid force is introduced. The performance function should be minimized satisfying the state equation. This problem can be transformed into the minimization problem without constraint condition by the Lagrange multiplier method and the adjoint equations using adjoint variables corresponding to the state equations. As a numerical study, the drag force minimization problem in the steady Stokes flow, which means approximated equation of the low Reynolds number Navier–Stokes equation is carried out. After that, the unsteady Navier–Stokes flow is analysed. As the minimization algorithm, the steepest descent method is successfully applied. Copyright © 2005 John Wiley & Sons, Ltd.     

Minimum drag shape in two-dimensional viscous flow

The problem of finding the shape of a body with smallest drag in a flow governed by the two-dimensional steady Navier-Stokes equations is considered. The flow is expressed in terms of a streamfunction which satisfies a fourth-order partial differential equation with the biharmonic operator as principal part. Using the adjoint variable approach, both the first- and second-order necessary conditions for the shape with smallest drag are obtained. An algorithm for the calculation of the optimal shape is proposed in which the first variations of solutions of the direct and adjoint problems are incorporated. Numerical examples show that the algorithm can produce the optimal shape successfully.     

Non-uniform suction control of flow around a circular cylinder

In the present study, numerical investigations were performed with optimisation to determine efficient non-uniform suction profiles to control the flow around a circular cylinder in the range of Reynolds numbers 4 < Re < 188.5. Several objectives were explored, namely the minimisation of the separation angle, total drag, and pressure drag. This was in an effort to determine the relationships between the characteristics of the uncontrolled flow and the parameters of the optimised suction control. A variety of non-uniform suction configurations were implemented and compared to the benchmark performance of uniform suction. It was determined that the best non-uniform suction profiles consisted of a distribution with a single locus and compact support. The centre of suction on the cylinder surface for the optimised control, and the quantity of suction necessary to achieve each objective, varied substantially with Reynolds number and also with the separation angle of the uncontrolled flows. These followed predictable relationships. Surprisingly, the location of optimised suction to eliminate separation did not follow the separation point as it moved with Re, but rather it moved in opposition to it towards the trailing edge of the cylinder. Non-uniform suction profiles were much more efficient at eliminating boundary layer separation, requiring the removal of less than half the volume of fluid as uniform control to achieve the same objective. Regardless of the method of control, less net suction was needed to minimise total drag than to eliminate separation, except at low Re. The results suggest that controlling the dynamic aspects of the flow has the most impact for reducing drag. This reinforces the usefulness of other studies that focus on the elimination of vortex shedding. The results show that the balance of drag components must be an important consideration when designing flow control systems and that, when done appropriately, substantial improvement can be seen in the flow characteristics.     

Comparison of adjoint‐based and feature‐based grid adaptation for functional outputs

Adjoint‐based and feature‐based grid adaptive strategies are compared for their robustness and effectiveness in improving the accuracy of functional outputs such as lift and drag coefficients. The output‐based adjoint approach strives to improve the adjoint error estimates that relate the local residual errors to the global error in an output function via adjoint variables as weight functions. A conservative adaptive indicator that takes into account the residual errors in both the primal (flow) and dual (adjoint) solutions is implemented for the adjoint approach. The physics‐based feature approach strives to identify and resolve significant features of the flow to improve functional accuracy. Adaptive indicators that represent expansions and compressions in the flow direction and gradients normal to the flow direction are implemented for the feature approach. The adaptive approaches are compared for functional outputs of three‐dimensional arbitrary Mach number flow simulations on mixed‐element unstructured meshes. Grid adaptation is performed with h‐refinement and results are presented for inviscid, laminar and turbulent flows. Copyright © 2006 John Wiley & Sons, Ltd.     

Optimal airfoil shapes for low Reynolds number flows

Flow over NACA 0012 airfoil is studied at α = 4^° and 12^° for Re?500. It is seen that the flow is very sensitive to Re. A continuous adjoint based method is formulated and implemented for the design of airfoils at low Reynolds numbers. The airfoil shape is parametrized with a non‐uniform rational B‐splines (NURBS). Optimization studies are carried out using different objective functions namely: (1) minimize drag, (2) maximize lift, (3) maximize lift to drag ratio, (4) minimize drag and maximize lift and (5) minimize drag at constant lift. The effect of Reynolds number and definition of the objective function on the optimization process is investigated. Very interesting shapes are discovered at low Re. It is found that, for the range of Re studied, none of the objective functions considered show a clear preference with respect to the maximum lift that can be achieved. The five objective functions result in fairly diverse geometries. With the addition of an inverse constraint on the volume of the airfoil the range of optimal shapes, produced by different objective functions, is smaller. The non‐monotonic behavior of the objective functions with respect to the design variables is demonstrated. The effect of the number of design parameters on the optimal shapes is studied. As expected, richer design space leads to geometries with better aerodynamic properties. This study demonstrates the need to consider several objective functions to achieve an optimal design when an algorithm that seeks local optima is used. Copyright © 2008 John Wiley & Sons, Ltd.     

Shape optimisation of an oscillating body in fluid flow by adjoint equation and ALE finite element methods

The purpose of this paper is to determine the shape of an oscillating body by minimising drag and lift forces, located in a transient incompressible viscous fluid flow by means of the Arbitrary Lagrangian Eulerian finite element method and an optimal control theory. A performance function is expressed by the drag and lift forces. The performance function should be minimised satisfying the state equation and the constant volume condition. Therefore, this problem can be transformed into a minimisation problem without constraint by the Lagrange multiplier method. The adjoint equation and the gradient of the performance function are used to update the shape of the body. In this study, as a minimisation technique, the weighted gradient method is applied. The final shape is obtained of which drag and lift forces are reduced by 66.2% and 92.8%, respectively. The final shape obtained by this study is compared with the final shape of the non-oscillating body. The obtained final shape of the oscillating body is significantly different from the non-oscillating body.     

Shape optimization of a body located in incompressible flow using adjoint method and partial control algorithm

The purpose of this study is to obtain an optimal shape of a body located in an incompressible viscous flow. The optimal shape of the body is defined so as to minimize the fluid forces acting on it by determining the surface coordinates based on the finite element method and the optimal control theory. The performance function, which is used to judge the optimality of a shape, is defined as the square sum of the drag and lift forces. The minimization problem is solved using an adjoint equation method. The gradient in the adjoint equation is affected by the finite element configuration. The use of a finite element mesh whose shape is appropriate for the procedure is important in shape optimization. If the finite element mesh used is not suitable for computations, the exact gradient is not calculated. Therefore, a structured mesh is used for the adjacent area of the body and all finite element meshes are refined using the Delaunay triangulation at each iteration computation. The weighted gradient method is applied as the minimization technique. Using an algorithm in which all nodal coordinates on the surface of the body are employed and starting from a circle as an initial shape, a front‐edged and rear‐round shape is obtained because of the vortices at the back of the body. To overcome this difficulty, we introduced the partial control algorithm, in which some of the nodal coordinates on the surface of the body are updated. From four cases of computational studies, we reveal that the optimal shape has both sharp front and sharp rear edges. All computations are conducted at Reynolds number Re=250. The minimum value of the performance function is obtained. Copyright © 2010 John Wiley & Sons, Ltd.     

On the generation of yawing moment using active flow control

The generation of control moments without moving control surfaces is of great practical importance. Following a successful flight demonstration of creating roll motion without ailerons using differential, lift oriented, flow control the current study is a first step towards generating yawing motion via differential flow controlled drag.A wind tunnel study was conducted on a 21% thick Glauert type airfoil. The upper surface flow is partially separated from the two-thirds chord location and downstream on this airfoil at all incidence angles. An array of mass-less Piezo-fluidic actuators, located at x/c = 0.65, are capable of fully reattaching the flow in a gradual, controlled manner. The actuators are individually operated such that the boundary layer could be controlled in a 3D fashion.Several concepts for creating yaw motion without moving control surface are examined. The ultimate goal is to generate the same lift on both wings, while decreasing the drag on one wing and increasing the drag on the other, therefore creating a yawing moment. Decreased drag is created by effective part-span separation delay while increased drag can be created by enhanced generation of vortex shedding or by highly localized 3D actuation.Detailed measurements of 3D surface pressure distributions and wake data with three velocity and streamwise vorticity components are presented and discussed along with surface flow visualization images. The data provide evidence that yawing moments can be generated with AFC.     

Adjoint‐based airfoil optimization with discretization error control

In this article, we develop a new airfoil shape optimization algorithm based on higher‐order adaptive DG methods with control of the discretization error. Each flow solution in the optimization loop is computed on a sequence of goal‐oriented h‐refined or hp‐refined meshes until the error estimation of the discretization error in a flow‐related target quantity (including the drag and lift coefficients) is below a prescribed tolerance. Discrete adjoint solutions are computed and employed for the multi‐target error estimation and adaptive mesh refinement. Furthermore, discrete adjoint solutions are employed for evaluating the gradients of the objective function used in the CGs optimization algorithm. Furthermore, an extension of the adjoint‐based gradient evaluation to the case of target lift flow computations is employed. The proposed algorithm is demonstrated on an inviscid transonic flow around the RAE2822, where the shape is optimized to minimize the drag at a given constant lift and airfoil thickness. The effect of the accuracy of the underlying flow solutions on the quality of the optimized airfoil shapes is investigated. Copyright © 2014 John Wiley & Sons, Ltd.     

An adjoint method for the calculation of remote sensitivities in supersonic flow

This paper presents an adjoint method for the calculation of remote sensitivities in supersonic flow. The goal is to develop a set of discrete adjoint equations and their corresponding boundary conditions in order to quantify the influence of geometry modifications on the pressure distribution at an arbitrary location within the domain of interest. First, this paper presents the complete formulation and discretization of the discrete adjoint equations. The special treatment of the adjoint boundary condition to obtain remote sensitivities or sensitivities of pressure distributions at points remotely located from the wing surface are discussed. Secondly, we present results that demonstrate the application of the theory to a three-dimensional remote inverse design problem using a low sweep biconvex wing and a highly swept blunt leading edge wing. Lastly, we present results that establish the added benefit of using an objective function that contains the sum of the remote inverse and drag minimization cost functions.     

A stabilized finite element method for shape optimization in low Reynolds number flows

A gradient‐based optimization procedure based on a continuous adjoint approach is formulated and implemented for steady low Reynolds number flows. A stabilized finite element formulation is proposed to solve the adjoint equations. The accuracy of the gradients from the adjoint approach is verified against the ones computed from a simple finite difference procedure. The validation of the formulation and its implementation is carried out via flow past an elliptical bump whose eccentricity is used as a design parameter. Shape design studies for the elliptical bump are then carried on with a more complex 4th order Bézier parametrization of the bump. Results for, both, optimal design and inverse problems are presented. Using different initial guesses, multiple optimal shapes are obtained. A multi‐objective function with additional constraints on the volume and the drag coefficient of the bump is utilized. It is seen that as more constraints are added to the objective function the design space is constrained and the multiple optimal shapes become progressively similar to each other. The study demonstrates the usefulness of this tool in obtaining multiple engineering solutions to a given design problem and also providing a framework to impose multiple constraints simultaneously. Copyright © 2007 John Wiley & Sons, Ltd.     

Suppression of force fluctuations in flow past an aerofoil

Force fluctuations on a solid body are associated with unsteadiness in the wake, e.g. vortex shedding. Therefore, the control of force fluctuations can be realised by suppressing the flow unsteadiness. A NACA0024 aerofoil closed with a round trailing edge is chosen to represent the solid body throughout this investigation, with the Reynolds number fixed at Re = 1000 and angle of attack α ≤ 15^o, at which the uncontrolled flow is two-dimensional. A linear optimal control is calculated by analysing the distribution of sensitivity of unsteadiness to control around the entire surface of the body. The nonlinear effects of the calculated control, which can be actuated through surface-normal suction and blowing across the surface of the aerofoil, are tested through two-dimensional direct numerical simulations. It is observed that a surface-normal velocity control with a maximum magnitude less than 8% of the free stream velocity completely suppresses unsteadiness at α = 10° with an overall drag reduction of 14% and a 138% increase of lift.     

Study of active flow control for a simplified vehicle model using the PANS method

Flow control has shown a potential in reducing the drag in vehicle aerodynamics. The present numerical study deals with active flow control for a quasi-2D simplified vehicle model using a synthetic jet (zero net mass flux jet). Recently developed near-wall Partially-Averaged Navier–Stokes (PANS) method, based on the ζ–f RANS turbulence model, is used. The aim is to validate the performance of this new method for the complex flow control problem. Results are compared with previous studies using LES and experiments, including global flow parameters of Strouhal number, drag coefficients and velocity profiles. The PANS method predicts a drag reduction of approximately 15%, which is closer to the experimental data than the previous LES results. The velocity profiles predicted by the PANS method agree well with LES results and experimental data for both natural and controlled cases. The PANS prediction showed that the near-wake region is locked-on due to the synthetic jet, and the shear layer instabilities are thus depressed which resulted in an elongated wake region and reduced drag. It demonstrates that the PANS method is able to predict the flow control problem well and is thus appropriate for flow control studies.     

Optimal control of shallow water flow using time-delay function

This paper presents an optimal control system that includes a time-delay function for application to flood control setups with a retardation area. This system consists of the present and past controls that express flow behaviour in the retardation area. Optimal control theory is used to obtain a control discharge that satisfies the state equation including the time-delay function and minimizes the performance function. The optimal control and the delayed control discharges are obtained by the solution of an adjoint equation. The weighted gradient method is employed as a minimization algorithm. The Galerkin finite element procedure is employed to discretize the state and adjoint equations in the spatial direction. The bubble function interpolation, originated by the authors' group, using a stabilized term, is employed for the discretization in space. The flood flow in the Tsurumi river is presented as a numerical model. We show in this paper that floods can be controlled by means of a time-delay function.     

Bifurcation and chaos of a flag in an inviscid flow

A two-dimensional model is developed to study the flutter instability of a flag immersed in an inviscid flow. Two dimensionless parameters governing the system are the structure-to-fluid mass ratio M^⁎ and the dimensionless incoming flow velocity U^⁎. A transition from a static steady state to a chaotic state is investigated at a fixed M^⁎=1 with increasing U^⁎. Five single-frequency periodic flapping states are identified along the route, including four symmetrical oscillation states and one asymmetrical oscillation state. For the symmetrical states, the oscillation frequency increases with the increase of U^⁎, and the drag force on the flag changes linearly with the Strouhal number. Chaotic states are observed when U^⁎ is relatively large. Three chaotic windows are observed along the route. In addition, the system transitions from one periodic state to another through either period-doubling bifurcations or quasi-periodic bifurcations, and it transitions from a periodic state to a chaotic state through quasi-periodic bifurcations.     

The turbulence vorticity as a window to the physics of friction-drag reduction by oscillatory wall motion

DNS data for channel flow, subjected to spanwise (in-plane) wall oscillations at a friction Reynolds number of 1025, are used to examine the turbulence interactions that cause the observed substantial reduction in drag provoked by the forcing. Following a review of pertinent interactions between the forcing-induced unsteady Stokes strain and the Reynolds stresses, identified in previous work by the present authors, attention is focused on the equations governing the components of the enstrophy, with particular emphasis placed on the wall-normal and the spanwise components. The specific objective is to study the mechanisms by which the Stokes strain modifies the enstrophy field, and thus the turbulent stresses. As such, the present analysis sheds fresh light on the drag-reduction processes, illuminating the interactions from a different perspective than that analysed in previous work. The investigation focuses on the periodic rise and fall in the drag and phase-averaged properties during the actuation cycle at sub-optimal actuation conditions, in which case the drag oscillates by around ±2% around the time-averaged 20% drag-reduction margin. The results bring out the important role played by specific strain-related production terms in the enstrophy-component equations, and also identifies vortex tilting/stretching in regions of high skewness as being responsible for the observed strong increase in the spanwise enstrophy components during the drag-reduction phase. Simultaneously, the wall-normal enstrophy component, closely associated with near-wall streak intensity, diminishes, mainly as a result of a reduction in a production term that involves the correlation between wall-normal vorticity fluctuations and the spanwise derivative of wall-normal-velocity fluctuations, which pre-multiplies the streamwise shear strain.     

Three-dimensional simulations of flow past two circular cylinders in side-by-side arrangements at right and oblique attacks

The flow past two identical circular cylinders in side-by-side arrangements at right and oblique attack angles is numerically investigated by solving the three-dimensional Navier–Stokes equations using the Petrov–Galerkin finite element method. The study is focused on the effect of flow attack angle and gap ratio between the two cylinders on the vortex shedding flow and the hydrodynamic forces of the cylinders. For an oblique flow attack angle, the Reynolds number based on the velocity component perpendicular to the cylinder span is defined as the normal Reynolds number Re_N and that based on the total velocity is defined as the total Reynolds number Re_T. Simulations are conducted for two Reynolds numbers of Re_N=500 and Re_T=500, two flow attack angles of α=0° and 45° and four gap ratios of G/D=0.5, 1, 3 and 5. The biased gap flow for G/D=0.5 and 1 and the flip-flopping bistable gap flow for G/D=1 are observed for both α=0° and 45°. For a constant normal Reynolds number of Re_N=500, the mean drag and lift coefficients at α=0° are very close to those at α=45°. The difference between the root mean square (RMS) lift coefficient at α=0° and that at α=45° is about 20% for large gap ratios of 3 and 5. From small gap ratios of 0.5 and 1, the RMS lift coefficients at α=0° and 45° are similar to each other. The present simulations show that the agreement in the force coefficients between the 0° and 45° flow attack angles for a constant normal Reynolds number is better than that for a constant total Reynolds number. This indicates that the normal Reynolds number should be used in the implementation of the independence principle (i.e., the independence of the force coefficients on the flow attack angle). The effect of Reynolds number on the bistable gap flow is investigated by simulating the flow for Re_N=100–600, α=0° and 45° and G/D=1. Flow for G/D=1 is found to be two-dimensional at Re_N=100 and weak three-dimensional at Re_N=200. While well defined biased flow can be identified for Re_N=300–600, the gap flow for Re_N=100 and 200 changes its biased direction too frequently to allow stable biased flow to develop.     

Adjoint optimization of natural convection problems: differentially heated cavity

Optimization of natural convection-driven flows may provide significant improvements to the performance of cooling devices, but a theoretical investigation of such flows has been rarely done. The present paper illustrates an efficient gradient-based optimization method for analyzing such systems. We consider numerically the natural convection-driven flow in a differentially heated cavity with three Prandtl numbers (\(Pr=0.15{-}7\)) at super-critical conditions. All results and implementations were done with the spectral element code Nek5000. The flow is analyzed using linear direct and adjoint computations about a nonlinear base flow, extracting in particular optimal initial conditions using power iteration and the solution of the full adjoint direct eigenproblem. The cost function for both temperature and velocity is based on the kinetic energy and the concept of entransy, which yields a quadratic functional. Results are presented as a function of Prandtl number, time horizons and weights between kinetic energy and entransy. In particular, it is shown that the maximum transient growth is achieved at time horizons on the order of 5 time units for all cases, whereas for larger time horizons the adjoint mode is recovered as optimal initial condition. For smaller time horizons, the influence of the weights leads either to a concentric temperature distribution or to an initial condition pattern that opposes the mean shear and grows according to the Orr mechanism. For specific cases, it could also been shown that the computation of optimal initial conditions leads to a degenerate problem, with a potential loss of symmetry. In these situations, it turns out that any initial condition lying in a specific span of the eigenfunctions will yield exactly the same transient amplification. As a consequence, the power iteration converges very slowly and fails to extract all possible optimal initial conditions. According to the authors’ knowledge, this behavior is illustrated here for the first time.     

Turbulence production in flow over a wavy wall

Measurements of the spatial and time variation of two components of the velocity have been made over a sinusoidal solid wavy boundary with a height to length ratio of 2a/λ = 0.10 and with a dimensionless wave number of α⁺ = (2π/λ)(v/u ^?) = 0.02. For these conditions, both intermittent and time-mean flow reversals are observed near the troughs of the waves. Statistical quantities that are determined are the mean streamwise and normal velocities, the root-meansquare of the fluctuations of the streamwise and normal velocities, and the Reynolds shear stresses. Turbulence production is calculated from these measurements. The flow is characterized by an outer flow and by an inner flow extending to a distance of about α^?1 from the mean level of the surface. Turbulence production in the inner region is fundamentally different from flow over a flat surface in that it is mainly associated with a shear layer that separates from the back of the wave. Flow close to the surface is best described by an interaction between the shear layer and the wall, which produces a retarded zone and a boundary-layer with large wall shear stresses. Measurements of the outer flow compare favorably with measurements over a flat wall if velocities are made dimensionless by a friction velocity defined with a shear stress obtained by extrapolating measurements of the Reynolds stress to the mean levels of the surface (rather than from the drag on the wall).