Numerical heat fluxcontrol forunsteady viscous fluid flows

This paper is devoted to the boundary control problems for non-stationary equations of viscous heat-conducting fluid. In these problems we change the fluid flow by heating or cooling some parts of the boundary. Numerical algorithm based on the finite dimensional minimization approach is proposed. Some computational results for vortex suppression in a cavity are presented and discussed.


Introduction
Optimization and control problems for viscous heat-conduction fluid flows have a large number of applications in science and mechanical engineering.There are numerous objectives of control, e.g., the flow or temperature field optimization, drag minimization, preventing transition to turbulence, vortex reduction, enhancing or deterring mixing.Usually these problems are formulated as minimization problems for suitable cost functionals and can be analyzed and solved by applying a unified approach based on the constrained optimization theory (see, for example, [1,2]).The same approach can be applied for inverse and parameter estimation problems.These problems for stationary flow models were studied in [3][4][5][6][7].An unsteady case is more complicated from theoretical and numerical points of view and requires the development of special numerical algorithms [8,9].
Numerical optimization for hydrodynamic mathematical models is connected with a number of difficulties.Minimization problems have some constraints because solutions must satisfy the governing equations.Usually a constrained minimization problem is rewritten as an unconstrained minimization problem using the Lagrange multipliers.Then one can formally calculate the derivatives of the Lagrangian and derive the necessary optimality conditions.However, a numerical solution of the optimality system for nonstationary equations is very difficult.Therefore we propose a new numerical algorithm to solve the control problem under consideration.This algorithm is based on the finite dimensional minimization approach where the control is considered as the linear combination of known basis functions with the unknown coefficients.The values of these coefficients are chosen in order to minimize some functional describing the objective of control.
A proposed algorithm is tested on the nonstationary control problem connected with the vortex suppression in the open cavity.We use the heat flux control on the cavity boundaries to obtain the fluid flow close to the potential flow with zero vorticity.

A mathematical model
Let Ω be a bounded two-dimensional domain with boundary .For simplicity, we assume that density ρ depends on temperature T as ρ=ρ*[1-β(T-T*)] according to the Boussines q approximation for the viscous fluid flow.Here ρ* and T* are the given reference density and temperature fields, is the thermal expansion coefficient.As a mathematical model we will consider the following initial boundary value problem: on the time interval (0, t max ).Here u, p and T denote the velocity, pressure and temperature respectively; is the kinematic viscosity coefficient, G is the gravitational acceleration vector, is the thermal diffusivity coefficient.We assume that boundary consists of four parts: inlet in , solid walls 0 , c and outlet out (Figure 1).On the inlet boundary section we prescribe the parabolic inflow profile for the velocity and the temperature of the fluid.We also consider no slip boundary condition u = 0 on the solid walls and 'do nothing' boundary condition on the outlet.The vertical boundary sections c will be used below for heat flux control.
The open cavity is the common case in many channels and industrial devices.The main problem of these cavities connected with the formation of vortices and stagnant zones.The main flow over the cavity is usually separated from the vortex in the cavity.Small particles can be accumulated in the cavity and in a circular motion would damage the wall surface like sandpaper.So sometimes you need to wash away the contents of the cavity.In many cases it is very difficult to install special injectors in the bottom.Therefore, we need a way to suppress the vortex and change the flow without permeation of the walls.If the fluid density is dependent on temperature, we can do it by means of the buoyancy force with heating and cooling of some boundary sections.In order to find an appropriate heat flux we need to solve the boundary control problem.

A numerical algorithm
A proposed algorithm is based on the main idea of paper [10] where an optimal boundary control problem for the stationary Navier-Stokes equations was solved using the finite dimensional minimization approach.Now we will extend this idea on the nonstationary case and a more complicated mathematical model.
At the beginning we split time interval (0, t max ) into N parts (t n-1 , t n ), n = 1, … , N. At each time step we shall simultaneously calculate heat flux boundary value and the corresponding solution of the initial boundary problem.Now we assume that we already know all values at time t=t n-1 and describe the calculation of values at time step t n .We shall consider heat flux (t n ) as a linear combination of known basis functions i with unknown coefficients k i , i = 1, … , S.Here k i are the new unknowns specific for this time step.
We will use a semidiscrete weak formulation for the initial boundary problem with linear implicit one-step approximation in time and finite element approximation in space.Therefore velocity u(t n ) can be written as follows: Where w n is the solution for homogeneous boundary condition = 0 while u j , j = 1, … , S are the solutions corresponding = j and homogeneous boundary conditions for velocity and temperature.
If we want velocityu(t n ) to be close to given field u d in some subdomain d then solution of the finite dimensional minimization problem for variables k i , i = 1, … , S can be found by solving the following system of linear algebraic equations: Here positive constant is a small regularization parameter.
The main steps of the algorithm are the following: find functions w n , u j ; calculate coefficient a ij , b j ; and solve the linear system for k j .Then we can find (t n ) and all other values on this time step.
Let us note that this algorithm does not use the first order necessary optimality conditions (see, for example, [1][2][3][4][5][6][7]) and is simpler to implement.It can be efficiently parallelized because S+1 boundary value problems to find w n and u j are solved independently.

Computational results
This numerical algorithm was applied for solving the vortex suppression problem in the open cavity (Figure 1).Fist of all, we find an uncontrolled flow solving the initial boundary value problem with homogeneous boundary condition = 0. We solve the non-dimensional equations with Prandtl number Pr = 7, Reynolds number Re = 10 and Grashof number Gr = 10000.At dimensionless time t=5 the flow becomes stable.The resulting velocity field in the cavity is shown in Figure 2.
It is clearly seen that most of the cavity is occupied by a large vortex.The main flow takes over only a small part and does not reach the bottom.We want to wash out this stagnant zone from the cavity.In order to do this we need to choosethe control, desired velocity field u d and solve the control problem.If we want to suppress the vortex then the desired flow is the potential flow with zero vorticity.Heat flux on vertical boundary sections c plays the role of control.
Then we solve the control problem using the algorithm described above.Let us note that this problem is essentially nonstationary.The flow changes gradually at each time step.At dimensionless time t = 5 the flow becomes stable.The resulting velocity field in the cavity can be seen in Figure 3.It is easy to notice that in this case we do not have a vortex.The main flow from the channel occupies the entire cavity.This flow can wash out all contents from the bottom.
The analysis of the temperature field shows that we have cooling on the left boundary and heating on the right boundary.Therefore, the cold fluid with a higher density goes down in the left part of the cavity and the heated fluid with a lower density moves upward in the right part of the cavity.
The main goal of the computational experiments was to determine the dependence of the solution accuracy on the choice of the problem parameters.We have considered different values of regularization parameter and the different number of basis functions S. The right choice of the control sections on the boundary also plays an important role.Based on the analysis of computational results we can choose optimal values of parameters and develop some recommendations for future applications.

Conclusion
We have considered control problems for nonstationary viscous heat-conducting fluid flows.A new numerical algorithm based on the finite dimensional minimization approach was proposed.This algorithm does not usethe Lagrange multipliersand the first order necessary optimality conditions.It has been successfully appliedto solve the problem of the vortex suppression in the open cavity.

Figure 2 .
Figure 2.An uncontrolled flow in the cavity.

Figure 3 .
Figure 3.A controlled flow in the cavity.
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