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]
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]
For a Newtonian fluid, the Cauchy stress tensor depends linearly on the components of the strain rate tensor:[3]
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:
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 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 functionss 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]
The weak formulation of Navier–Stokes equations reads:[3]
Find, for all , , such that
with , where[3]
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]
Find, for all , , such that
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
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
Let's define the finite element (FE) spaces of continuous functions, (polynomials of degree on each element of the triangulation) as[3]
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]
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]
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]
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]
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
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]
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
- ^ 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.
- ^ 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.
- ^ 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.
- ^ a b c d e Pope, Stephen B. (2000). Turbulent Flows by Stephen B. Pope. Cambridge University Press. ISBN 9780521598866.
- ^ a b Quarteroni, Alfio; Sacco, Riccardo; Saleri, Fausto (2007). Numerical Mathematics (2 ed.). Springer-Verlag. ISBN 9783540346586.
- ^ 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.
- ^ 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.
- ^ 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.
- ^ 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.