We consider the problem of an appropriate choice of a cost functional for vortex reduction for unsteady flows described by the Navier–Stokes equations. This choice is directly related to a physically correct definition of a vortex. Therefore, we discuss different possibilities for the cost functional and analyze the resulting optimal control problems. Moreover, we present an efficient numerical realization of this concept based on space-time finite element discretization and demonstrate its behavior in some numerical experiments. It is demonstrated that the choice of cost functionals has a significant effect on the reduction of vortices.

MSC codes

  1. 35Q30
  2. 76D05
  3. 76D55
  4. 48J20


  1. optimal control
  2. vortex reduction
  3. Navier–Stokes equations

Get full access to this article

View all available purchase options and get full access to this article.


F. Abergel and R. Temam, On some control problems in fluid mechanics, Theoret. Comput. Fluid Dynamics, 1 (1990), pp. 303–325.
R. Becker and M. Braack, A finite element pressure gradient stabilization for the Stokes equations based on local projections, Calcolo, 38 (2001), pp. 137–199.
R. Becker, D. Meidner, and B. Vexler, Efficient numerical solution of parabolic optimization problems by finite element methods, Optim. Methods Softw., to appear.
R. Becker and B. Vexler, Optimal control of the convection-diffusion equation using stabilized finite element methods, Numer. Math., 106 (2007), pp. 349–367.
M. Berggren, R. Glowinski, and J.-L. Lions, A computational approach to controllability issues for flow-related models. (I): Pointwise control of the viscous Burgers equation, Int. J. Comput. Fluid Dyn., 7 (1996), pp. 237–252.
H. M. Blackburn, N. N. Mansour, and B. J. Cantwell, Topology of fine-scale motions in turbulent channel flow, J. Fluid Mech., 310 (1996), pp. 293–324.
M. S. Chong, A. E. Perry, and B. J. Cantwell, A general classification of three-dimensional flow fields, Phys. Fluids A, 2 (1990), pp. 765–777.
F. H. Clarke, Optimization and Nonsmooth Analysis, John Wiley and Sons, New York, 1983.
R. Dautray and J.-L. Lions, Mathematical Analysis and Numerical Methods for Science and Technology, Vol. 6, Springer Verlag, Berlin, 1993.
K. Eriksson, Cl. Johnson, and V. Thomée, Time discretization of parabolic problems by the discontinuous Galerkin method, RAIRO Model. Math. Anal. Numer., 19 (1985), pp. 611–643.
A. V. Fursikov, M. D. Gunzburger, and L. S. Hou, Boundary value problems and optimal boundary control for the Navier–Stokes system: The two-dimensional case, SIAM J. Control Optim., 36 (1998), pp. 852-894.
M. Giles, M. Larson, M. Levenstam, and E. Suli, Adaptive Error Control for Finite Element Approximations of the Lift and Drag Coefficients in Viscous Flow, Technical report NA-76/06, Oxford University Computing Laboratory, Oxford, UK, 1997.
V. Girault and P.-A. Raviert, Finite Element Methods for Navier–Stokes Equations, Springer-Verlag, Berlin, 1979.
A. Griewank, Achieving logarithmic growth of temporal and spatial complexity in reverse automatic differentiation, Optim. Methods Softw., 1 (1992), pp. 35–54.
M. Gunzburger, ed., Flow Control, IMA Vol. Math. Appl. 68, Springer, Berlin, 1995.
G. Haller, Lagrangian structures and the rate of strain in a partition of two-dimensional turbulence, Phys. Fluids, 13 (2001), pp. 3365–3385.
G. Haller, An objective definition of a vortex, J. Fluid Mech., 525 (2005), pp. 1–26.
A. Haraux, How to differentiate the projection on a closed convex set in Hilbert space. Some applications to variational inequalities, J. Math. Soc. Japan, 29 (1977), pp. 615–631.
J. G. Heywood, R. Rannacher, and S. Turek, Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations, Internat. J. Numer. Methods Fluids, 22 (1996), pp. 325–352.
M. Hintermüller, K. Kunisch, Y. Spasov, and S. Volkwein, Dynamical systems based optimal control of incompressible fluids, Internat. J. Numer. Methods Fluids, 46 (2004), pp. 345–359.
M. Hinze and K. Kunisch, Second order methods for boundary control of the instationary Navier–Stokes system, ZAMM Z. Angew. Math. Mech., 84 (2004), pp. 171–187.
J. C. R. Hunt, A. A. Wray. and P. Moin, Eddies, Stream and Convergence Zones in Turbulent Flows, Report CTR-S88, Center for Turbulence Research, Stanford University, Stanford, CA, 1988.
J. Jeong and F. Hussain, On the identificaton of vortex, J. Fluid Mech., 285 (1995), pp. 69–94.
K. Kunisch and B. Vexler, Constrained Dirichlet boundary control in $L^2$ for a class of evolution equations, SIAM J. Control Optim., to appear.
G. Lapeyre, B. L. Hua, and P. Klein, Dynamics of the orientation of active and passive scalars in two-dimensional turbulence, Phys. Fluids, 13 (2001), pp. 251–264.
G. Lapeyre, P. Klein, and B. L. Hua, Does the tracer gradient vector align with the strain eigenvectors in 2D turbulence?, Phys. Fluids, 11 (1999), pp. 3729–3737.
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, Heidelberg, 1971.
H. J. Lugt, The dilemma of defining a vortex, in Recent Developments in the Theoretical and Experimental Fluid Mechanics, U. Müller, K. G. Roesner, and B. Schmidt, eds., Springer, Berlin, 1979, pp. 309–321.
J. Nocedal and S. J. Wright, Numerical optimization, Springer Ser. Oper. Res., Springer, New York, 1999.
A. Okubo, Horizontal dispersion of floatable trajectories in the vicinity of velocity singularities such as convergencies, Deep-Sea Res., 17 (1970), pp. 445–454.
A. Rösch and B. Vexler, Optimal control of the Stokes equations: A priori error analysis for finite element discretization with postprocessing, SIAM J. Numer. Anal., 44 (2006), pp. 1903–1920.
V. A. Solonnikov, On the differential properties of the solution of the first boundary value problem for nonstationary systems of Navier–Stokes equations, Trudy Math. Inst. Steklov, 73 (1964), pp. 222–291.
V. A. Solonnikov, Estimates in $L^P$ of solutions of elliptic and parabolic systems, Proc. Steklov Inst. Math., 102 (1967), pp. 157–185.
M. Tabor and I. Klapper, Stretching and alignment in chaotic and turbulent flows, Chaos Solitons Fractals, 4 (1994), pp. 1031–1055.
R. Temam, Navier–Stokes Equations. Theory and Numerical Analysis, North–Holland, Amsterdam, 1984.
F. Tröltzsch and D. Wachsmuth, Second-order sufficient optimality conditions for the control of Navier Stokes equations, ESAIM Control Optim. Calc. Var., 12 (2006), pp. 93–119.
B. Vexler, Finite element approximation of elliptic dirichlet optimal control problems, Numer. Funct. Anal. Optim., to appear.
J. Weiss, The dynamics of enstrophy transfer in two-dimensional hydrodynamics, Phys. D, 48 (1991), pp. 273–294.

Information & Authors


Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 1368 - 1397
ISSN (online): 1095-7138


Submitted: 31 May 2005
Accepted: 26 February 2007
Published online: 14 September 2007

MSC codes

  1. 35Q30
  2. 76D05
  3. 76D55
  4. 48J20


  1. optimal control
  2. vortex reduction
  3. Navier–Stokes equations



Metrics & Citations



If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By







Copy the content Link

Share with email

Email a colleague

Share on social media

The SIAM Publications Library now uses SIAM Single Sign-On for individuals. If you do not have existing SIAM credentials, create your SIAM account https://my.siam.org.