Optimal Control of Thermally Convected Fluid Flows.
We examine the optimal control of stationary thermally convected fluid flows from the theoretical and numerical point of view.  We use thermal convection as control mechanism; that is, control is effected through the temperature on part of the boundary.  Control problems are formulated as constrained minimization problems. Existence of optimal control is given and a first-order necessary condition of optimality from which optimal solutions can be obtained is established. We develop numerical methods to solve the necessary condition of optimality and present computational results for control of cavity- and channel-type flows showing the feasibility of the proposed approach.
Introduction
. The control of viscous flows for the purpose of achieving some desired
objective is crucial to many technological and scientific applications. In the past,
these control problems have been addressed either through expensive experimental processes
or through the introduction of significant simplifications into the analyses used in
the development of control mechanisms. Recently mathematicians and scientists have been
able to address flow control problems in a systematic, rigorous manner and established a
mathematical and numerical foundation for these problems; see [1-2], [4-5], [8-9], [11], and
[15-18].
This work was supported in part by the Air Force Office of Scientific Research under grants AFOSR
F49620-95-1-0437 and AFOSR F49620-95-1-0447.
y Center for Research in Scientific Computation, Department of Mathematics, North Carolina State
University, Raleigh, NC 27695-8205 (kito@eos.ncsu.edu and ravi@eos.ncsu.edu).
K. ITO AND S.S. RAVINDRAN
The control of vorticity has significant applications in science and engineering such as
control of turbulence and control of crystal growth process. In this article we consider
the minimization of vorticity in viscous incompressible thermally convected flows using
boundary temperature as control mechanism.
We formulate the control problem as a constrained optimization problem for steady
viscous incompressible thermally convected flow, namely that of computing a boundary
temperature on a part of the boundary that minimizes the vorticity in the fluid. The
constraint is the system of equations that represents steady viscous incompressible Navier-Stokes
equations coupled with the energy equation. The choice for the cost is a quadratic
functional involving the vorticity in the fluid so that a minimum of that functional corresponds
to the minimum possible vorticity subject to the constraints. We then prove the
existence of an optimal control and derive the first-order necessary conditions characterizing
the control. Once the necessary optimality conditions are derived, we develop numerical
methods to solve such conditions and present numerical results showing the feasibility of
the approach for cavity and channel type flows.
1.1. The governing equations of a thermally convected flow. The class of thermally
convective flow we consider is modelled by Boussinesq equations whose derivation is
based on certain assumptions about the thermodynamics and the thermal effects on the
flow. The first one is that variations in density is negligible except for the body force term
aeg in the momentum equations, where ae is the density and the vector g is the constant
acceleration of gravity. We next assume that the density ae in the term aeg can be given by
are reference temperature and density, respectively,
T is the absolute temperature and fi is the thermal expansion coefficient. Furthermore, we
assume that in the energy equation, the dissipation of mechanical energy is negligible and
the viscosity -, the heat conductivity -, the thermal expansion coefficient fi and the specific
heat at constant pressure c p are constant. Then under these assumptions the steady flow
is governed by following equations:
where\Omega is a bounded open set and the heat source is assumed to be zero. If we assume
there is a length scale ', a velocity scale U and a temperature scale in the flow,
then one can define nondimensional Prandtl number
Reynolds number Re
according to x / x=', u
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 3
obtain the following nondimensional form of Boussinesq equations.
Re \Deltau
Re 2
where g is now a unit vector in the direction of gravitational acceleration.
1.2. Statement of the optimal control problem. Let us next state the optimal
control problem we consider
\Omega jr \Theta uj 2
subject to the state
Re
Re 2
ReP r
with the boundary conditions as follows. are
disjoint portions of the boundary \Gamma of the
domain\Omega\Gamma
where are given on the boundary and g is a temperature control by the
radiational heating or cooling. In the cost functional J , the term
R\Omega
jr \Theta uj 2
d\Omega is a measure
of vorticity \Theta u in the flow, the term R
is the measure of the magnitude
of the control which is also required for the rigorous mathematical analysis of the control
problem and the penalizing parameter ffi adjusts the size of the terms in the cost. The flow
quantities u, T and p denote as usual the velocity, temperature and pressure, respectively.
The outline of the paper is as follows. In x2, we give a variational formulation of the
state equations and study their wellposedness. We believe it is new since it deals with
nonhomogeneous boundary conditions. In x3 the existence of optimal solutions and first
order optimality conditions for optimal control problems are established. x4 deals with
computational methods to solve the necessary conditions of optimality. Finally, in x5, we
present numerical results for control of cavity and channel flows using boundary temperature
controls.
4 K. ITO AND S.S. RAVINDRAN
1.3. Notations. Throughout, C or C i (where i is any subscript) denotes a constant
depending only the
domain\Omega which is assumed to be a bounded set in R
I 2 with smooth
boundary \Gamma. We denote by L
2(\Omega\Gamma the collection of square-integrable functions defined on
Z
\Omega q
@x ff 1@x ff 22 L
Vector-valued counterparts of these spaces are denoted by bold-face symbols, e.g., H
are the restriction to the boundary of H
We denote the norms and inner products for H
tively. The L 2
or L
inner product is denoted by (\Delta; \Delta). We denote the norms and
inner products for H r (\Gamma) or H r (\Gamma) by k \Delta k r;\Gamma and (\Delta; \Delta) r;\Gamma , respectively. The L 2 (\Gamma) or L 2 (\Gamma)
inner product is denoted by (\Delta; \Delta) \Gamma .
be the divergence free subspace of H 1
defined by
and H 0 is the completion of V 0 with respect to L
norm and is given by
The space H 0 is equipped with the norm k \Delta k 0 and V 0 is equipped with juj
1 be the subspace of H
defined by
and set
1 be the strong dual spaces of V 0 and V 1 , respectively, and
h\Delta; \Deltai denote the dual product on either V
Throughout the mathematical
discussions, for the sake of convenience we set b
Re
ReP r
and b
Re 2
which are not
to be confused with the physical quantities such as kinematic viscosity and conductivity.
We define the following bilinear and trilinear forms
a 0 (u;
Z\Omega
Z\Omega
Z\Omega
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 5
Z
and
Z
We have the coercivity relations associated with a 0 (\Delta; \Delta) and a 1 (\Delta; \Delta):
a 0 (u;
and
which are a direct consequence of Poincar'e inequality.
2. Weak Formulation. In this section we discuss the weak variational formulation
of the Boussinesq system (1.2) and establish the existence of weak solutions.
It follows from the Hopf extension (see [10]) that for each ffl ? 0, there exists a function
(\Omega\Gamma such that r \Delta
provided that the boundary data u
In the sequel we
will take
. Let T 2 H
1(\Omega\Gamma be a function such that T
any function (u; T
satisfying the inhomogeneous boundary condition
can be represented by
We then obtain a weak variational form of (1.2). For
a
A solution (u; is called a weak solution of (1.2) if equation (2.1) is
Regarding the bilinear form b 0 (\Delta; \Delta; \Delta), we have the following results.
Lemma 2.1. For
1(\Omega\Gamma , the trilinear form b 0 (\Delta; \Delta; \Delta) satisfies
6 K. ITO AND S.S. RAVINDRAN
Proof. The first inequality follows from the Holders inequality. We obtain
The second result follows from Green's formula
provided that r \Delta
It follows from the proof of Lemma 2.1 that
and
for
2.1. Wellposedness. In this section we prove the existence of a weak solution to (2.1).
Theorem 2.2. Given exists a weak solution (u; to (2.1) and
Moreover, if g(x), T 0 (x) and T 1 (x) are bounded below by T 1 and bounded above by T 2
almost everywhere then almost everywhere
in\Omega for every solution.
Proof. Step I (Existence): We show that (2.1) has a solution (u; Given
linear equations by
(2:6a) a 0 (u; v)
where
First, we show that (2.6) has a unique
solution (w; we show that the solution map S on V 0 +u defined by S(b
is the unique solution to (2.6), has a fixed point by Sch-auder fixed point
theorem. The fixed point the corresponding solution T
solution to (2.1).
We first note, from Lemma 2.1 and (2.5), that the bilinear form oe 1 (\Delta; \Delta) defined by
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 7
on It thus follows from Lax-Milgram theorem that the
equation
for
Setting in (2.6b), we have
It follows from (2.5) that b 1 (b u; /;
a 1 (/; /)+2 b -h k/k 2
This implies
Similarly, letting
From the definition of j and /, it follows that kTk 0 - C 6 which is independent of b
Next, we define the bilinear form oe 0 (\Delta; \Delta) on V 0 \Theta V 0 by
It then follows from Lemma 2.1, (2.4) and the inequality
that oe 0 (\Delta; \Delta) is bounded and V 0 -coercive. Thus, by Lax-Milgram theorem, the equation
for has a unique solution w
in (2.6a) and using the estimate kTk 0 - C 6 , we get
be a closed convex subspace of H
1(\Omega\Gamma2 defined by
Then it follows from (2.7) that S maps from \Sigma\Sigma \Sigma into \Sigma\Sigma \Sigma . Moreover, the solution map S is
compact. In fact, if b
converges weakly to b
8 K. ITO AND S.S. RAVINDRAN
is compactly embedded into
be the corresponding
solution of (2.6), respectively to b u k and b u. Then we have
for Setting we have from Lemma 2.1 and (2.5) that
which implies kT Similarly, we have
and thus ju by Sch-auder fixed point theorem (see [20]) there exists at
least one solution to (2.6).
Let us next derive the appriori estimate. Setting in (2.6b) we obtain
a
0:
Equivalently,
a 1 (';
Then using the coercivity and continuity properties of a 1 (\Delta; \Delta) and b 1 (\Delta; \Delta; \Delta) and the antisymmetry
property of b 1 (\Delta; \Delta; \Delta), it follows that
for some constant C 11 independent of b
u. From (2.7)-(2.8), we obtain the appriori estimate
for some constant C.
Step II (L 1 estimate): We show that if T 1 -
almost everywhere x
for every solution (u; to (2.1). In fact, letting in the second
equation of (2.1) and using the same arguments as above, we obtain
where
Thus, we obtain k/k 2
Similarly, one can prove
that choosing the test function
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 9
We also have the uniqueness of solutions under the smallness assumption on u and
Theorem 2.3. If g(x), T 0 (x) and T 1 (x) are bounded below by T 1 and bounded above
by T 2 almost everywhere and if jT are sufficiently small, then (2.1) has a
unique solution in Z.
Proof. Suppose are two solutions to (2.1). Then letting e
and e
a
a
for (v; /) 2 V. Setting
T , we obtain, using (2.4) and (2.5), that
a
and
a
This implies
(b
and
Hence if
then je uj
From Theorem 2.2 and (2.7), we have
Thus, if jT sufficiently small then (2.1) has a unique solution in Z.
K. ITO AND S.S. RAVINDRAN
3. Existence of Optimal Controls and Necessary Optimality Conditions. In
this section, we show the existence of optimal solutions for the minimization problem (1.1)-
and estabilish a necessary optimality condition. Let us first assume that C is a closed
convex subset of L 2 (\Gamma 1 ). For example C can be defined to be
almost everywhere g
or us denote the set
Let us define the cost functional J (u; T ; g) to be
and cast the control problems in the following abstract setting: For
Z \Theta C with
Minimize J (x)
subject to
where the equality constraint represents the state equations (2.1),
(b ff Tg; v)
for (v; /) 2 V, where
Then, we have the existence of solutions to the optimal control problem.
Theorem 3.1. Consider the minimization problem:
Minimize
where C is a closed convex subset of L 2 (\Gamma 1 ). Assume that the function
I
is convex and lower semicontinuous and satisfies " '(z) - c 1 kzk 2
I
the minimization problem has a solution.
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 11
Proof. Let C be a minimizing sequence. Since
is uniformly bounded in k and thus from (2.9) so is k(u Hence there exists a
subsequence of fkg, which will be denoted by the same index, such that
weakly to (u; T ; g) 2 Z \Theta C, since V \Theta L 2 space and C is a closed and convex
set. Since H 1
(\Omega\Gamma is compactly embedded into
4(\Omega\Gamma0 it follows from Lemma 2.1 that
and
which implies (u; T ) 2 S(g). Now, since " ' is convex and lower semicontinuous it follows
from [3] that (u; T ; g) minimizes (3.1).
Assume that x  denotes an optimal pair of (3.1). Then we
have the following theorem.
Theorem 3.2. Assume that x   is a regular point in the sense that
Then there exists Lagrange multipliers (ii such that
for (v; /) 2 V and
Proof. It follows from [14] that if (3.2) is satisfied, then there exists a Lagrange multipliers
(ii such that
for all (v; that is
for all (v; /) 2 V and j 2 C. Setting (v;
(3.5), we obtain (3.3).
Concerning the regular point condition (3.2), we have
12 K. ITO AND S.S. RAVINDRAN
Lemma 3.3. If g   2 int (C) then the regular point condition (3.2) is equivalent to the
following condition. Suppose
Proof. If g   2 int (C) then (3.2) is equivalent to the
linear is a unique solution to
Then, since H 1
(\Omega\Gamma is embedded compactly to L
4(\Omega\Gamma2 by Lemma 2.1, F is compact. Thus,
it follows from Banach closed range and Riesz-Schauder theorems that E 0
surjective if and only if ker(G   which is equivalent to (3.6).
Finally, if and the cost functional is given as in (1.1) then (3.3)-(3.4) can
be equivalently written as
a 0 (ii
ff (ii
for all (v; /) 2 V.
To facilitate the computational discussion, let us collect the necessary conditions of
optimality (2.1) and (3.7) and recast them by using the vector decomposition of L
(see [10]), L 2
and by introducing pressure p and adjoint
pressure -. We obtain: For
a
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 13
a 0 (ii
+(r \Theta u; r \Theta
c(ii
ff (ii
The system (3.8)-(3.9) forms the necessary conditions of optimality that optimal states and
control must satisfy. This system will also be called the optimality system.
4. Computational Methods. In this section we describe a computational method
to find the optimal control by solving the optimality system (3.8)-(3.9).
4.1. Finite Element Approximations. A finite element discretization of the optimality
system (3.8)-(3.9) is defined in the usual manner. First one chooses families of
finite dimensional subspaces X h ae H
1(\Omega\Gamma and S h ae L
2(\Omega\Gamma4 We let X 0
These families are parameterized by a parameter h
that tends to zero; commonly, h is chosen to be some measure of the grid size. These finite
dimensional function spaces are defined on an approximate
domain\Omega h . For simplicity we
will state our results in this section by
We assume that these finite
element spaces satisfy the following approximation properties (see, [7, 10]): there exist an
integer k and a constant C, independent of h, v, q and /, such that
and
Here we may choose any pair of subspaces X h and S h such that X 0
h and S 0
h can be used
for finding finite element approximations of solutions of the Navier-Stokes equations with
homogeneous Dirichlet conditions. Thus, we make the following standard assumptions,
which are exactly those employed in well-known finite element methods for the Navier-Stokes
equations. Next, we assume the inf-sup condition: there exists a constant C, independent
of h, such that
sup
R\Omega
d\Omega
14 K. ITO AND S.S. RAVINDRAN
This condition assures the stability of finite element discretizations of the Navier-Stokes
equations and also that of the optimlity system (3.8)-(3.9). The references [9] and [6] may
also be consulted for a catalogue of finite element subspaces that meet the requirements
of the above approximation properties and the inf-sup condition. We also define Z h to be
Once the approximating subspaces have been chosen, we look for an approximate optimal
solution
h \Theta X h \Theta X 0
h \Theta S 0
h \Theta X h \Theta Z h by solving
the discrete optimality system of equations
a
a
+(r \Theta u
We next briefly sketch the proof of optimal error estimates. We first prove optimal error
estimates for the approximations of the linearized optimality system. Then by a careful
choice of spaces and operators we can fit the optimality system into the framework of Brezzi-
Rappaz-Raviart theory (see [10]). By verifying all the requirements of that theory, we obtain
optimal error estimates for the approximation of the optimality system of equations.
Theorem 4.1. Assume
(\Omega\Gamma is a nonsingular solution of the optimality system (3.8)-(3.9). Then for
each sufficiently small h, the approximate optimality system (3.10) has a unique solution
h \Theta X h \Theta X 0
h \Theta S 0
h \Theta X h in a neighborhood of (u;
such that
We employ Newton's iteration method to solve this finite dimensional nonlinear system
of equations.
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 15
4.2. Newtons Method. The Newton's method based on exact Jacobian for solving
the discrete optimality system is given as follows:
Triangulate the flow domain with a sufficiently small mesh size h; choose finite element
spaces X h and S h ; choose an initial guess
n ) from the following discrete system
of equations:
a
a
+(r \Theta u
At each Newton's iteration, we solve the linear system of equations by Gaussian eliminations
for banded matrices. Under suitable assumptions, Newton's method converges at
a quadratic rate to the finite element solution convergence
of Newton's method is valid within a contraction ball. In practice we normally
first perform a few successive approximations and then switch to the Newton's method.
The successive approximations are defined by replacing the second, fourth, fifth and sixth
equations in the Newton's iterations by
a
a
+(r \Theta u
In the case of the uncontrolled Navier-Stokes equations, the solution is unique for small
Reynolds numbers and the successive approximations converge globally and linearly; see
K. ITO AND S.S. RAVINDRAN
[7]. However, in the present case of an optimal system of equations for the Navier-Stokes
equations, the solution is not shown to be unique and the successive approximation is not
shown to be globally convergent, even for small Reynolds numbers. Our numerical experience
seems to suggest that the global convergence of the successive approximations for
the optimality system is still valid for small Reynolds numbers. Thus the combined successive
approximations-Newton iterations gives an effective method for solving the discrete
optimality system of equations.
5. Computational Results. We will consider two test examples for vorticity minimization
using boundary temperature control. Both examples are related to optimization
and control of vapuor transport process for crystal growth. Some related works are reported
in [19], [13] and [5]. In [19], tracking temperature field in an ampoule using boundary
temperature control is considered, tracking a desired history of the freezing interface
location/motion in conduction driven solidification process using temperature control is
considered [13] and some optimal control problems in combustion are discussed in [5].
5.1. Numerical Example 1. In this example, we consider the control of vorticity
in a backward-facing-step channel flow. The vorticity is caused by the injection of flow
at the inlet of the channel and we try to control the vorticity or the recirculation rather
by adjusting the temperature at the top and bottom walls. A schematic of the backward-
facing-step channel is shown in Figure 1. The height of the step is 0.5 and that of the outflow
boundary is 1. The length of the very bottom of the channel is 5 and the total horizontal
length is 6. Figure 3 demonstrate the flow situation for high Reynolds numbers which is
computed with and the following boundary conditions for velocity and
temperature.
@n
@n
@n
The parameters were taken as follows: Re = 200,
The computational domain is divided into around 350 triangles with refined grid near the
corner, see Figure 2. The finite element spaces X h and X h are chosen to be piecewise
quadratic elements (for u h and T h ) defined over triangles and the space S h is chosen to be
piecewise linear element (for p h ) defined over the same triangles.
A recirculation appears at the corner region whose size increases with increasing Reynolds
number. The objective is to shape the recirculation region by applying temperature control
on the very top boundary \Gamma top and bottom boundary \Gamma bottom .
We take the corner region of the
Figure 1, for vorticity
minimization. The control is computed by solving the optimality system (3.9)-(3.10) by
applying finite element and Newtons method described in x4.1-2. The parameter in the
functional was chosen as and the adjoint state variables ii i, - and - were discretized
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 17
using the same way as their state counterparts. At each Newton iteration a banded Gaussian
elimination was used to solve the resulting linear system. We obtain the optimal solution
typically in 7 Newton iterations.

Figure

4 gives the controlled velocity field u h , Figures 5 and 6 are the blow-up of the
uncontrolled and controlled flows, respectively, at the corner of the backward-facing-step.

Figures

7 and 8 are the control distributions on the top and bottom boundaries.
The values of the integral
R\Omega
jr \Theta uj 2
d\Omega without and with controls were .94 and .51,
respectively. We see that we achieved a reduction of 45.74% in the L
2(\Omega\Gamma2671/ of the
vorticity.
5.2. Numerical Example 2. This example is motivated by the transport process in
high pressure chemical vapour deposition (CVD) reactors (see [10-11] and [6]). A typical
vertical reactor, shown in Figure 9, is a classical configuration for the growth of compound
semiconductors by metalorganic vapor phase epitaxy. The reactant gases are introduced at
the top of the reactor and flow down to the substrate (\Gamma 2 ) which is kept at high temperature.
This means that least dense gas is closest to the substrate and the flow is likely to be
affected by buoyancy driven convection. In order to have uniform growth rates and better
compositional variations, it is essential to have flow field without recirculations.
Our objective here is to minimize the vorticity by adjusting the temperature at the side
walls in order to obtain a flow field without recirculations and thereby obtain better
vertical transport.
The geometry of the prototype reactor, depicted in Figure 9, has two outlet portions,
and an inlet, \Gamma i , whose widths are 1/3. The size of the susceptor region \Gamma 2 and that of
the side walls \Gamma 1 are 1; the height of the inlet port \Gamma s is 1/3.
The boundary conditions for computations were as follows:
For the uncontrolled flow computations, we take throughout the computations
in this problem we take the Reynolds number to be Re = 100, the Prandtl number to be
1. For the discretization, the finite element spaces were chosen to be the
same as in the previous example.
We performed simulations with several values of Gr=Re 2 for the uncontrolled case.
The flow situations are shown in Figure 10a)-15a) and the corresponding vorticity in L 2
norm is given in Table-I. Two standing circulation appear near the susceptor due to natural
convection which did not appear at all when Gr=Re 2 - 1. For the control simulations
heating/cooling control was applied to the side walls \Gamma 1 with fixed inflow rate and vorticity
cost was minimized with the parameter 0:01. This control problem was solved using
our optimal control techniques.
K. ITO AND S.S. RAVINDRAN
The resulting flow fields for various Gr=Re 2 values are shown in Figure 10b)-15b) and
the corresponding vorticity in L 2 norm is given in Table-I. The control values on the side
walls are given in Figure 16a)-b). We see, in Figure 10b)-15b), significant reduction in
recirculation for the controlled flow. Our computational experiments (not reported here)
indicate that for Gr=Re 2 AE 1, thermal control mechanism on the side walls with fixed flow
rates may be less effective for the elimination of recirculation.
Uncontrolled Vorticity 0.1983 0.2600 0.3505 0.4712 0.6186 0.7878
Controlled Vorticity 0.1126 0.1174 0.1113 0.1123 0.1147 0.1183

Table

I. Uncontrolled and Controlled Vorticity in L 2 norm for different Gr=Re 2
6. Conclusion. In this article we studied vorticity minimization problem in fluid flows
using boundary temperature controls. We formulated the problem as constrained minimization
problem with cost functional being the vorticity in the flow. We proved the existence
of optimal solution and the existence of Lagrange multipliers. The necessary conditions
of optimality was given characterizing the controls and optimal states. Newton's method
combined with mixed finite element method is used to solve the necessary conditions of
optimality. We finally solved two canonical problems demonstrating the feasibility of the
approach.
OPTIMAL CONTROL OF THERMALLY CONVECTED FLUID FLOWS 19



--R


Optimal Control of Navier-Stokes Equations
Analysis of Convex Variational Problems
Existence of Optimal Controls for Viscous Flow Problems

Transport Phenomena in Vertical Reactors for Metalor- ganic Vapor Phase Epitaxy
Finite Element Methods for Viscous Incompressible Flows
Analysis and Finite Element Approximation of Optimal Control Problems for the Stationary Navier-Stokes Equations with Dirichlet Controls

Finite Element Methods for Navier-Stokes Equations
Optimal Control of Thermally coupled Navier-Stokes equations
Mathematical Issues in Optimal Design of a Vapour Transport Reactors
Control of the Freezing Interface Motion in Two-dimensional Solidification Process Using the Adjoint Method
First and Second Order Necessary and Sufficient Optimality Conditions for Infinite-dimensional Programming Problems
Numerical Solutions of Optimal Boundary Control Problems for Navier-Stokes Flows
An Optimal Control Problem in Exterior Hydrodynamics


Optimal Design of Nonlinear Parabolic Systems.
New York
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flow control;finite element methods;temperature control;optimization;navier-stokes equations