Streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations

The streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations can be used for finite element computations of high Reynolds number incompressible flow using equal order of finite element space (i.e. ) by introducing additional stabilization terms in the Navier–Stokes Galerkin formulation.[1][2]

The finite element (FE) numerical computation of incompressible Navier–Stokes equations (NS) suffers from two main sources of numerical instabilities arising from the associated Galerkin problem.[1] Equal order finite elements for pressure and velocity, (for example, ), do not satisfy the inf-sup condition and leads to instability on the discrete pressure (also called spurious pressure).[2] Moreover, the advection term in the Navier–Stokes equations can produce oscillations in the velocity field (also called spurious velocity).[2] Such spurious velocity oscillations become more evident for advection-dominated (i.e., high Reynolds number ) flows.[2] To control instabilities arising from inf-sup condition and convection dominated problem, pressure-stabilizing Petrov–Galerkin (PSPG) stabilization along with Streamline-Upwind Petrov-Galerkin (SUPG) stabilization can be added to the NS Galerkin formulation.[1][2]

The incompressible Navier–Stokes equations for a Newtonian fluid

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Let   be the spatial fluid domain with a smooth boundary  , where   with   the subset of   in which the essential (Dirichlet) boundary conditions are set, while   the portion of the boundary where natural (Neumann) boundary conditions have been considered. Moreover,  , and  . Introducing an unknown velocity field   and an unknown pressure field  , in absence of body forces, the incompressible Navier–Stokes (NS) equations read[3]   where   is the outward directed unit normal vector to  ,   is the Cauchy stress tensor,   is the fluid density , and   and   are the usual gradient and divergence operators. The functions   and   indicate suitable Dirichlet and Neumann data, respectively, while   is the known initial field solution at time  .

For a Newtonian fluid, the Cauchy stress tensor   depends linearly on the components of the strain rate tensor:[3]   where   is the dynamic viscosity of the fluid (taken to be a known constant) and   is the second order identity tensor, while   is the strain rate tensor  

The first of the NS equations represents the balance of the momentum and the second one the conservation of the mass, also called continuity equation (or incompressible constraint).[3] Vectorial functions  ,  , and   are assigned.

Hence, the strong formulation of the incompressible Navier–Stokes equations for a constant density, Newtonian and homogeneous fluid can be written as:[3]

Find,  , velocity   and pressure   such that:   where,   is the kinematic viscosity, and   is the pressure rescaled by density (however, for the sake of clearness, the hat on pressure variable will be neglect in what follows).

In the NS equations, the Reynolds number shows how important is the non linear term,  , compared to the dissipative term,   [4]  

The Reynolds number is a measure of the ratio between the advection convection terms, generated by inertial forces in the flow velocity, and the diffusion term specific of fluid viscous forces.[4] Thus,   can be used to discriminate between advection-convection dominated flow and diffusion dominated one.[4] Namely:

  • for "low"  , viscous forces dominate and we are in the viscous fluid situation (also named Laminar Flow),[4]
  • for "high"  , inertial forces prevail and a slightly viscous fluid with high velocity emerges (also named Turbulent Flow).[4]

The weak formulation of the Navier–Stokes equations

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The weak formulation of the strong formulation of the NS equations is obtained by multiplying the first two NS equations by test functions   and  , respectively, belonging to suitable function spaces, and integrating these equation all over the fluid domain  .[3] As a consequence:[3]  

By summing up the two equations and performing integration by parts for pressure ( ) and viscous ( ) term:[3]  

Regarding the choice of the function spaces, it's enough that   and  ,   and  , and their derivative,   and   are square-integrable functions in order to have sense in the integrals that appear in the above formulation.[3] Hence,[3]  

Having specified the function spaces  ,   and  , and by applying the boundary conditions, the boundary terms can be rewritten as[3]   where  . The integral terms with   vanish because  , while the term on   become  

The weak formulation of Navier–Stokes equations reads:[3]

Find, for all  ,  , such that  

with  , where[3]  

Finite element Galerkin formulation of Navier–Stokes equations

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In order to numerically solve the NS problem, first the discretization of the weak formulation is performed.[3] Consider a triangulation  , composed by tetrahedra  , with   (where   is the total number of tetrahedra), of the domain   and   is the characteristic length of the element of the triangulation.[3]

Introducing two families of finite-dimensional sub-spaces   and  , approximations of   and   respectively, and depending on a discretization parameter  , with   and  ,[3]   the discretized-in-space Galerkin problem of the weak NS equation reads:[3]

Find, for all  ,  , such that   with  , where   is the approximation (for example, its interpolant) of  , and  

Time discretization of discretized-in-space NS Galerkin problem can be performed, for example, by using the second order Backward Differentiation Formula (BDF2), that is an implicit second order multistep method.[5] Divide uniformly the finite time interval   into   time step of size  [3]  

For a general function  , denoted by   as the approximation of  . Thus, the BDF2 approximation of the time derivative reads as follows:[3]  

So, the fully discretized in time and space NS Galerkin problem is:[3]

Find, for  ,  , such that   with  , and   is a quantity that will be detailed later in this section.

The main issue of a fully implicit method for the NS Galerkin formulation is that the resulting problem is still non linear, due to the convective term,  .[3] Indeed, if   is put, this choice leads to solve a non-linear system (for example, by means of the Newton or Fixed point algorithm) with a huge computational cost.[3] In order to reduce this cost, it is possible to use a semi-implicit approach with a second order extrapolation for the velocity,  , in the convective term:[3]  

Finite element formulation and the INF-SUP condition

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Let's define the finite element (FE) spaces of continuous functions,   (polynomials of degree   on each element   of the triangulation) as[3]   where,   is the space of polynomials of degree less than or equal to  .

Introduce the finite element formulation, as a specific Galerkin problem, and choose   and   as[3]  

The FE spaces   and   need to satisfy the inf-sup condition(or LBB):[6]  

with  , and independent of the mesh size  [6] This property is necessary for the well posedness of the discrete problem and the optimal convergence of the method.[6] Examples of FE spaces satisfying the inf-sup condition are the so named Taylor-Hood pair   (with  ), where it can be noticed that the velocity space   has to be, in some sense, "richer" in comparison to the pressure space  [6] Indeed, the inf-sup condition couples the space   and  , and it is a sort of compatibility condition between the velocity and pressure spaces.[6]

The equal order finite elements,   ( ), do not satisfy the inf-sup condition and leads to instability on the discrete pressure (also called spurious pressure).[6] However,   can still be used with additional stabilization terms such as Streamline Upwind Petrov-Galerkin with a Pressure-Stabilizing Petrov-Galerkin term (SUPG-PSPG).[2][1]

In order to derive the FE algebraic formulation of the fully discretized Galerkin NS problem, it is necessary to introduce two basis for the discrete spaces   and  [3]   in order to expand our variables as[3]  

The coefficients,   ( ) and   ( ) are called degrees of freedom (d.o.f.) of the finite element for the velocity and pressure field, respectively. The dimension of the FE spaces,   and  , is the number of d.o.f, of the velocity and pressure field, respectively. Hence, the total number of d.o.f   is  .[3]

Since the fully discretized Galerkin problem holds for all elements of the space   and  , then it is valid also for the basis.[3] Hence, choosing these basis functions as test functions in the fully discretized NS Galerkin problem, and using bilinearity of   and  , and trilinearity of  , the following linear system is obtained:[3]   where   ,  ,  ,  , and   are given by[3]   and   and   are the unknown vectors[3]  

Problem is completed by an initial condition on the velocity  . Moreover, using the semi-implicit treatment  , the trilinear term   becomes bilinear, and the corresponding matrix is[3]  

Hence, the linear system can be written in a single monolithic matrix ( , also called monolithic NS matrix) of the form[3]   where  .

Streamline upwind Petrov–Galerkin formulation for incompressible Navier–Stokes equations

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NS equations with finite element formulation suffer from two source of numerical instability, due to the fact that:

  • NS is a convection dominated problem, which means "large"  , where numerical oscillations in the velocity field can occur (spurious velocity);
  • FE spaces   are unstable combinations of velocity and pressure finite element spaces, that do not satisfy the inf-sup condition, and generates numerical oscillations in the pressure field (spurious pressure).

To control instabilities arising from inf-sup condition and convection dominated problem, Pressure-Stabilizing Petrov–Galerkin(PSPG) stabilization along with Streamline-Upwind Petrov–Galerkin (SUPG) stabilization can be added to the NS Galerkin formulation.[1]

  where   is a positive constant,   is a stabilization parameter,   is a generic tetrahedron belonging to the finite elements partitioned domain  ,   is the residual of the NS equations.[1]

  and   is the skew-symmetric part of the NS equations[1]  

The skew-symmetric part of a generic operator   is the one for which  [5]

Since it is based on the residual of the NS equations, the SUPG-PSPG is a strongly consistent stabilization method.[1]

The discretized finite element Galerkin formulation with SUPG-PSPG stabilization can be written as:[1]

Find, for all    , such that   with  , where[1]  

and  , and   are two stabilization parameters for the momentum and the continuity NS equations, respectively. In addition, the notation   has been introduced, and   was defined in agreement with the semi-implicit treatment of the convective term.[1]

In the previous expression of  , the term   is the Brezzi-Pitkaranta stabilization for the inf-sup, while the term   corresponds to the streamline diffusion term stabilization for large  .[1] The other terms occur to obtain a strongly consistent stabilization.[1]

Regarding the choice of the stabilization parameters  , and  :[2]  

where:   is a constant obtained by an inverse inequality relation (and   is the order of the chosen pair  );   is a constant equal to the order of the time discretization;   is the time step;   is the "element length" (e.g. the element diameter) of a generic tetrahedra belonging to the partitioned domain  .[7] The parameters   and   can be obtained by a multidimensional generalization of the optimal value introduced in[8] for the one-dimensional case.[9]

Notice that the terms added by the SUPG-PSPG stabilization can be explicitly written as follows[2]  

 

where, for the sake of clearness, the sum over the tetrahedra was omitted: all the terms to be intended as  ; moreover, the indices   in   refer to the position of the corresponding term in the monolithic NS matrix,  , and   distinguishes the different terms inside each block[2]  

Hence, the NS monolithic system with the SUPG-PSPG stabilization becomes[2]   where  , and  .

It is well known that SUPG-PSPG stabilization does not exhibit excessive numerical diffusion if at least second-order velocity elements and first-order pressure elements ( ) are used.[8]

References

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  1. ^ a b c d e f g h i j k l m Tezduyar, T. E. (1 January 1991). "Stabilized Finite Element Formulations for Incompressible Flow Computations††This research was sponsored by NASA-Johnson Space Center (under grant NAG 9-449), NSF (under grant MSM-8796352), U.S. Army (under contract DAAL03-89-C-0038), and the University of Paris VI". Advances in Applied Mechanics. 28. Elsevier: 1–44. doi:10.1016/S0065-2156(08)70153-4.
  2. ^ a b c d e f g h i j Tobiska, Lutz; Lube, Gert (1 December 1991). "A modified streamline diffusion method for solving the stationary Navier–Stokes equation". Numerische Mathematik. 59 (1): 13–29. doi:10.1007/BF01385768. ISSN 0945-3245. S2CID 123397636.
  3. ^ a b c d e f g h i j k l m n o p q r s t u v w x y z aa ab ac ad ae af ag Quarteroni, Alfio (2014). Numerical Models for Differential Problems (2 ed.). Springer-Verlag. ISBN 9788847058835.
  4. ^ a b c d e Pope, Stephen B. (2000). Turbulent Flows by Stephen B. Pope. Cambridge University Press. ISBN 9780521598866.
  5. ^ a b Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2007). Numerical Mathematics (2 ed.). Springer-Verlag. ISBN 9783540346586.
  6. ^ a b c d e f Brezzi, Franco; Fortin, Michel (1991). Mixed and Hybrid Finite Element Methods (PDF). Springer Series in Computational Mathematics. Vol. 15. doi:10.1007/978-1-4612-3172-1. ISBN 978-1-4612-7824-5.
  7. ^ Forti, Davide; Dedè, Luca (August 2015). "Semi-implicit BDF time discretization of the Navier–Stokes equations with VMS-LES modeling in a High Performance Computing framework". Computers & Fluids. 117: 168–182. doi:10.1016/j.compfluid.2015.05.011.
  8. ^ a b Shih, Rompin; Ray, S. E.; Mittal, Sanjay; Tezduyar, T. E. (1992). "Incompressible flow computations with stabilized bilinear and linear equal-order-interpolation velocity-pressure elements". Computer Methods in Applied Mechanics and Engineering. 95 (2): 221. Bibcode:1992CMAME..95..221T. doi:10.1016/0045-7825(92)90141-6. S2CID 31236394.
  9. ^ Kler, Pablo A.; Dalcin, Lisandro D.; Paz, Rodrigo R.; Tezduyar, Tayfun E. (1 February 2013). "SUPG and discontinuity-capturing methods for coupled fluid mechanics and electrochemical transport problems". Computational Mechanics. 51 (2): 171–185. Bibcode:2013CompM..51..171K. doi:10.1007/s00466-012-0712-z. hdl:11336/1065. ISSN 1432-0924. S2CID 123650035.