Variational multiscale method

Consider an open bounded domain $\backslash Omega\; \backslash subset\; \backslash mathbb\; R^d$ with smooth boundary $\backslash Gamma\; \backslash subset\; \backslash mathbb\; R^\{d-1\}$, being $d\; \backslash geq\; 1$ the number of space dimensions. Denoting with $\backslash mathcal\; L$ a generic, second order, nonsymmetric differential operator, consider the following boundary value problem:

:$$

\text{find } u: \Omega \to \mathbb R \text{ such that}:

:$$

\begin{cases}

\mathcal L u = f & \text{ in } \Omega \\

u = g & \text{ on } \Gamma\\

\end{cases}

being $f:\; \backslash Omega\; \backslash to\; \backslash mathbb\; R$ and $g:\; \backslash Gamma\; \backslash to\; \backslash mathbb\; R$ given functions. Let $H^1\; (\backslash Omega)$ be the Hilbert space of square-integrable functions with square-integrable derivatives:

:$$

H^1(\Omega)= \{ f \in L^2(\Omega): \nabla f \in L^2 (\Omega)\}.

Consider the trial solution space $\backslash mathcal\; V\_g$ and the weighting function space $\backslash mathcal\; V$ defined as follows:

:$$

\mathcal V_g = \{u \in H^1 (\Omega): \, u=g \text{ on } \Gamma \},

:$$

\mathcal V = H_0^1(\Omega) = \{v \in H^1 (\Omega): \, v=0 \text{ on } \Gamma \}.

The variational formulation of the boundary value problem defined above reads:

:$\backslash text\{find\; \}\; u\; \backslash in\; \backslash mathcal\; V\_g\; \backslash text\{\; such\; that:\; \}\; a(v,\; u)\; =\; f(v)\; \backslash ,\; \backslash ,\; \backslash ,\; \backslash ,\; \backslash forall\; v\; \backslash in\; \backslash mathcal\; V$,

being $a(v,\; u)$ the bilinear form satisfying $a(v,\; u)\; =\; (v,\; \backslash mathcal\; L\; u)$, $f(v)=(v,\; f)$ a bounded linear functional on $\backslash mathcal\; V$ and $(\backslash cdot,\; \backslash cdot)$ is the $L^2(\backslash Omega)$ inner product. Furthermore, the dual operator $\backslash mathcal\; L^*$ of $\backslash mathcal\; L$ is defined as that differential operator such that $\backslash mathcal\; (v,\; \backslash mathcal\; L\; u)\; =\; (\backslash mathcal\; L^*v,\; u)\backslash ,\; \backslash ,\; \backslash ,\; \backslash forall\; u,\; \backslash ,\; v\; \backslash in\; \backslash mathcal\; V$.

File:VMS figure.png

In VMS approach, the function spaces are decomposed through a multiscale direct sum decomposition for both $\backslash mathcal\; V\_g$ and $\backslash mathcal\; V$ into coarse and fine scales subspaces as:

:$$

\mathcal V= \bar {\mathcal V} \oplus \mathcal V'

and

:$$

\mathcal V_g= \bar {\mathcal V_g} \oplus \mathcal V_g'.

Hence, an overlapping sum decomposition is assumed for both $u$ and $v$ as:

:$u\; =\; \backslash bar\{u\}\; +\; u\text{'}\; \backslash text\{\; and\; \}\; v\; =\; \backslash bar\{v\}\; +\; v\text{'}$,

where $\backslash bar\; u$ represents the coarse (resolvable) scales and $u\text{'}$ the fine (subgrid) scales, with $\backslash bar\{u\}\; \backslash in\; \backslash bar\{\backslash mathcal\; V\_g\}$, $\{u\text{'}\}\; \backslash in\; \{\backslash mathcal\; V\_g\}\text{'}$, $\backslash bar\{v\}\; \backslash in\; \backslash bar\{\backslash mathcal\; V\}$ and $v\text{'}\; \backslash in\; \{\backslash mathcal\; V\}\text{'}$. In particular, the following assumptions are made on these functions:

:$$

\begin{align}

\bar u = g & & \text{ on } \Gamma & & \forall& \bar u \in \bar{\mathcal{V_g}}, \\

u' = 0 & & \text{ on } \Gamma & & \forall& u' \in {\mathcal{V_g}}', \\

\bar v = 0 & & \text{ on } \Gamma & & \forall& \bar v \in \bar{\mathcal{V}}, \\

v' = 0 & & \text{ on } \Gamma & & \forall& v' \in {\mathcal{V}}'.

\end{align}

With this in mind, the variational form can be rewritten as

:$$

a(\bar v + v', \bar u + u') = f (\bar v + v')

and, by using bilinearity of $a\; (\backslash cdot\; ,\; \backslash cdot)$ and linearity of $f\; (\backslash cdot)$,

:$$

a (\bar v, \bar u) + a (\bar v, u') + a (v', \bar u) + a(v', u') = f(\bar v) + f (v').

Last equation, yields to a coarse scale and a fine scale problem:

:$$

\text{find } \bar u \in \bar{\mathcal V}_g \text{ and } u' \in \mathcal V' \text{ such that: }

:$$

\begin{align}

& & a (\bar v, \bar u) + a (\bar v, u') &= f(\bar v) & & \forall \bar v \in \bar{\mathcal V} & \, \, \, \, \text{coarse-scale problem}\\

& & a (v', \bar u) + a (v', u') &= f( v') & & \forall v' \in {\mathcal V}' & \, \, \, \, \text{fine-scale problem} \\

\end{align}

or, equivalently, considering that $a(v,\; u)\; =\; (v,\; \backslash mathcal\; L\; u)$ and $f(v)=(v,\; f)$:

:$$

\text{find } \bar u \in \bar{\mathcal V}_g \text{ and } u' \in \mathcal V' \text{ such that: }

:$$

\begin{align}

& & (\bar v, \mathcal L \bar u) + (\bar v, \mathcal L u') &= (\bar v, f) & & \forall \bar v \in \bar{\mathcal V}, \\

& & (v', \mathcal L \bar u) + (v', \mathcal L u') &= ( v', f) & & \forall v' \in {\mathcal V}'.\\

\end{align}

By rearranging the second problem as $(v\text{'},\; \backslash mathcal\; L\; u\text{'})\; =\; -\; (v\text{'},\; \backslash mathcal\; L\; \backslash bar\; u\; -\; f)$, the corresponding Euler–Lagrange equation reads:

:$$

\begin{cases}

\mathcal L u' = - (\mathcal L \bar u - f) & \text{ in } \Omega \\

u' = 0 & \text{ on } \Gamma

\end{cases}

which shows that the fine scale solution $u\text{'}$ depends on the strong residual of the coarse scale equation $\backslash mathcal\; L\; \backslash bar\; u\; -\; f$.{{cite journal |last1=Hughes |first1=Thomas J.R. |title=Multiscale phenomena: Green's functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods |journal=Computer Methods in Applied Mechanics and Engineering |date=November 1995 |volume=127 |issue=1–4 |pages=387–401 |doi=10.1016/0045-7825(95)00844-9|bibcode=1995CMAME.127..387H |doi-access=free }} The fine scale solution can be expressed in terms of $\backslash mathcal\; L\; \backslash bar\; u\; -\; f$ through the Green's function $G:\; \backslash Omega\; \backslash times\; \backslash Omega\; \backslash to\; \backslash mathbb\; R\; \backslash text\{\; with\; \}\; G=0\; \backslash text\{\; on\; \}\; \backslash Gamma\; \backslash times\; \backslash Gamma$:

:$$

u'(y) = - \int_\Omega G(x, y) (\mathcal L \bar u - f )(x)\,d \Omega_x \, \, \, \forall y \in \Omega.

Let $\backslash delta$ be the Dirac delta function, by definition, the Green's function is found by solving $\backslash forall\; y\; \backslash in\; \backslash Omega$

:$$

\begin{cases}

\mathcal L^* G(x, y) = \delta (x-y) & \text{ in } \Omega \\

G(x,y)=0 & \text{ on } \Gamma

\end{cases}

Moreover, it is possible to express $u\text{'}$ in terms of a new differential operator $\backslash mathcal\; M$ that approximates the differential operator $-\; \backslash mathcal\; L^\{-1\}$ as

:

u' = \mathcal M (\mathcal L \bar u - f),

with $\backslash mathcal\; M\; \backslash approx\; -\; \backslash mathcal\; L^\{-1\}$. In order to eliminate the explicit dependence in the coarse scale equation of the sub-grid scale terms, considering the definition of the dual operator, the last expression can be substituted in the second term of the coarse scale equation:

:$$

(\bar v, \mathcal L u') = (\mathcal L^* \bar v, u') = (\mathcal L^* \bar v, \mathcal M (\mathcal L \bar u - f)).

Since $\backslash mathcal\; M$ is an approximation of $-\; \backslash mathcal\; L^\{-1\}$, the Variational Multiscale Formulation will consist in finding an approximate solution $\backslash tilde\{\backslash bar\; u\}\; \backslash approx\; \backslash bar\; u$ instead of $\backslash bar\; u$. The coarse problem is therefore rewritten as:

:$$

\text{find } \tilde{\bar u} \in \mathcal{\bar V}_g: \; \; \; a (\bar v, \tilde{\bar u}) + (\mathcal L^* \bar v, \mathcal M (\mathcal L \tilde{\bar u} - f)) = (\bar v, f) \; \; \; \forall \bar{v} \in \mathcal {\bar V},

being

:$$

(\mathcal L^* \bar v, \mathcal M (\mathcal L \tilde{\bar u} - f)) = - \int_{\Omega} \int_{\Omega} (\mathcal L^* \bar v)(y) G(x, y) (\mathcal L \tilde{\bar u} - f)(x) \,d \Omega_x \,d\Omega_y.

Introducing the form

:$$

B(\bar v, \tilde{\bar u}, G) = a (\bar v, \tilde{\bar u}) + (\mathcal L^* \bar v, \mathcal M (\mathcal L \tilde{\bar u}))

and the functional

:$$

L(\bar v, G)= (\bar v, f) + (\mathcal L^* \bar v, \mathcal M f)

,

the VMS formulation of the coarse scale equation is rearranged as:

:$$

\text{find } \tilde{\bar u} \in \mathcal{\bar V}_g: \, B(\bar v, \tilde{\bar u}, G) = L(\bar v, G) \, \, \, \forall \bar{v} \in \mathcal {\bar V}.

Since commonly it is not possible to determine both $\backslash mathcal\; M$ and $G$, one usually adopt an approximation. In this sense, the coarse scale spaces $\backslash bar\{\backslash mathcal\; V\}\_g$ and $\backslash bar\{\backslash mathcal\; V\}$ are chosen as finite dimensional space of functions as:

:$$

\bar{\mathcal V}_g \equiv \mathcal V_{g_h} : = \mathcal V_g \cap X_r^h(\Omega)

and

:$$

\bar{\mathcal V} \equiv \mathcal V_{h} : = \mathcal V \cap X_h^r(\Omega),

being $X\_r^h(\backslash Omega)$ the Finite Element space of Lagrangian polynomials of degree $r\; \backslash geq\; 1$ over the mesh built in $\backslash Omega$ .{{cite book |author-link1= Alfio Quarteroni |last1=Quarteroni |first1=Alfio |title=Numerical models for differential problems |publisher=Springer |isbn=978-3-319-49316-9 |edition=Third |date=2017-10-10 }} Note that $\backslash mathcal\{V\}\_g\text{'}$ and $\backslash mathcal\{V\}\text{'}$ are infinite-dimensional spaces, while $\backslash mathcal\{V\}\_\{g\_h\}$ and $\backslash mathcal\{V\}\_h$ are finite-dimensional spaces.

Let $u\_h\; \backslash in\; \backslash mathcal\; V\_\{g\_h\}$ and $v\_h\; \backslash in\; \backslash mathcal\; V\_\{h\}$ be respectively approximations of $\backslash tilde\{\backslash bar\; u\}$ and $\{\backslash bar\; v\}$, and let $\backslash tilde\; G$ and $\backslash tilde\{\backslash mathcal\; M\}$ be respectively approximations of $G$ and $\{\backslash mathcal\; M\}$. The VMS problem with Finite Element approximation reads:

:$$

\text{find } u_h \in \mathcal V_{g_h}: B(v_h, u_h, \tilde G) = L( v_h, \tilde G) \, \, \, \forall {v}_h \in \mathcal { V}_h

or, equivalently:

:$$

\text{find } u_h \in \mathcal V_{g_h}: a (v_h, u_h) + (\mathcal L^* v_h, \mathcal {\tilde{M}} (\mathcal L { u_h} - f)) = ( v_h, f) \, \, \, \forall {v}_h \in \mathcal { V}_h

Consider an advection–diffusion problem:

:$$

\begin{cases}

-\mu \Delta u + \boldsymbol b \cdot \nabla u = f & \text{ in } \Omega \\

u=0 & \text{ on } \partial \Omega

\end{cases}

where $\backslash mu\; \backslash in\; \backslash mathbb\; R$ is the diffusion coefficient with $\backslash mu>0$ and $\backslash boldsymbol\; b\; \backslash in\; \backslash mathbb\; R^d$ is a given advection field. Let $\backslash mathcal\{V\}=\; H^1\_0(\backslash Omega)$ and $u\; \backslash in\; \backslash mathcal\; V$, $\backslash boldsymbol\; b\; \backslash in\; [L^2(\backslash Omega)]^d$, $f\; \backslash in\; L^2(\backslash Omega)$. Let $\backslash mathcal\; L\; =\; \backslash mathcal\; L\_\{diff\}\; +\; \backslash mathcal\; L\_\{adv\}$, being $\backslash mathcal\; L\_\{diff\}\; =\; -\; \backslash mu\; \backslash Delta$ and $\backslash mathcal\; L\_\{adv\}\; =\; \backslash boldsymbol\; b\; \backslash cdot\; \backslash nabla$.

The variational form of the problem above reads:

:$$

\text{find} \, u \in \mathcal V: \; \; \; a(v, u) = (f, v) \; \; \; \forall v \in \mathcal V,

being

:$$

a(v, u) = (\nabla v, \mu \nabla u) + (v, \boldsymbol b \cdot \nabla u).

Consider a Finite Element approximation in space of the problem above by introducing the space $\backslash mathcal\; V\_h\; =\; \backslash mathcal\; V\; \backslash cap\; X\_h^r$ over a grid $\backslash Omega\_h\; =\; \backslash bigcup\_\{k=1\}^\{N\}\; \backslash Omega\_k$ made of $N$ elements, with $u\_h\; \backslash in\; \backslash mathcal\; V\_h$.

The standard Galerkin formulation of this problem reads

:$$

\text{find } u_h \in \mathcal V_h: \; \; \; a(v_h, u_h) = (f, v_h) \; \; \; \forall v \in \mathcal V,

Consider a strongly consistent stabilization method of the problem above in a finite element framework:

:$$

\text{ find } u_h \in \mathcal V_h: \, \, \, a(v_h, u_h) + \mathcal L_h (u_h, f; v_h)= (f, v_h) \, \, \, \forall v_h \in \mathcal V_h

for a suitable form $\backslash mathcal\; L\_h$ that satisfies:

:$$

\mathcal L_h (u, f; v_h) = 0 \, \, \, \forall v_h \in \mathcal V_h.

The form $\backslash mathcal\; L\_h$ can be expressed as $(\backslash mathbb\; L\; v\_h,\; \backslash tau\; (\backslash mathcal\; L\; u\_h\; -\; f))\_\{\backslash Omega\_h\}$, being $\backslash mathbb\; L$ a differential operator such as:

:$$

\mathbb L=

\begin{cases}

+ \mathcal L & \, \, \, & \text{ Galerkin/least squares (GLS)} \\

+ \mathcal L_{adv} & \, \, \, & \text{ Streamline Upwind Petrov-Galerkin (SUPG)} \\

- \mathcal L^* & \, \, \, & \text{ Multiscale} \\

\end{cases}

and $\backslash tau$ is the stabilization parameter. A stabilized method with $\backslash mathbb\; L\; =\; -\backslash mathcal\; L^*$ is typically referred to multiscale stabilized method . In 1995, Thomas J.R. Hughes showed that a stabilized method of multiscale type can be viewed as a sub-grid scale model where the stabilization parameter is equal to

:$$

\tau= - \tilde{\mathcal M} \approx - \mathcal M

or, in terms of the Green's function as

:$$

\tau \delta (x-y) = \tilde G(x, y) \approx G(x,y),

which yields the following definition of $\backslash tau$:

:$$

\tau = \frac{1}

For the 1-d advection diffusion problem, with an appropriate choice of basis functions and $\backslash tau$, VMS provides a projection in the approximation space.{{cite journal |last1= Hughes|first1= T.J.|last2= Sangalli|first2= G.|title= Variational multiscale analysis: the fine-scale Green's function, projection, optimization, localization, and stabilized methods|journal= SIAM Journal on Numerical Analysis|date= 2007|volume= 45|issue= 2|pages= 539–557|publisher= SIAM |doi= 10.1137/050645646}} Further, an adjoint-based expression for $\backslash tau$ can be derived,{{cite journal |last1= Garg|first1= V.V.|last2= Stogner |first2= R.|title= Local enhancement of functional evaluation and adjoint error estimation for variational multiscale formulations|journal= Computer Methods in Applied Mechanics and Engineering|date= 2019|volume= 354|pages= 119–142| publisher= Elsevier|doi= 10.1016/j.cma.2019.05.023}}

:$$

\tau_e = -\frac{\mathcal L(\tilde{z}, u_h)_e}{(\phi_{e}L_h(u_h)),L^*(\tilde{z}))_e}

where $\backslash tau\_e$ is the element wise stabilization parameter, $\backslash mathcal\; L(\backslash tilde\{z\},\; u\_h)\_e$ is the element wise residual and the adjoint $\backslash tilde\{z\}$ problem solves,

:$$

\mathcal a(\tilde{z}, v) + L_h(\tilde{z}, v) = \int_{\Omega_e} v \, dx

In fact, one can show that the $\backslash tau$ thus calculated allows one to compute the linear functional $\backslash int\_\{\backslash Omega\}\; u\; \backslash ,\; dx$ exactly.

The idea of VMS turbulence modeling for Large Eddy Simulations(LES) of incompressible Navier–Stokes equations was introduced by Hughes et al. in 2000 and the main idea was to use - instead of classical filtered techniques - variational projections.{{cite journal |last1=Hughes |first1=Thomas J.R. |last2=Mazzei |first2=Luca |last3=Jansen |first3=Kenneth E. |title=Large Eddy Simulation and the variational multiscale method |journal=Computing and Visualization in Science |date=May 2000 |volume=3 |issue=1–2 |pages=47–59 |doi=10.1007/s007910050051|s2cid=120207183 }}{{cite journal |last1=Bazilevs |first1=Y. |last2=Calo |first2=V.M. |last3=Cottrell |first3=J.A. |last4=Hughes |first4=T.J.R. |last5=Reali |first5=A. |last6=Scovazzi |first6=G. |title=Variational multiscale residual-based turbulence modeling for large eddy simulation of incompressible flows |journal=Computer Methods in Applied Mechanics and Engineering |date=December 2007 |volume=197 |issue=1–4 |pages=173–201 |doi=10.1016/j.cma.2007.07.016|bibcode=2007CMAME.197..173B }}

Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density $\backslash rho$ in a domain $\backslash Omega\; \backslash in\; \backslash mathbb\; R^d$ with boundary $\backslash partial\; \backslash Omega\; =\; \backslash Gamma\_D\; \backslash cup\; \backslash Gamma\_N$, being $\backslash Gamma\_D$ and $\backslash Gamma\_N$ portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied ($\backslash Gamma\_D\; \backslash cap\; \backslash Gamma\_N\; =\; \backslash emptyset$):

:$$

\begin{cases}

\rho \dfrac{\partial \boldsymbol u}{\partial t} + \rho (\boldsymbol u \cdot \nabla) \boldsymbol u - \nabla \cdot \boldsymbol \sigma (\boldsymbol u, p) = \boldsymbol f & \text{ in } \Omega \times (0, T) \\

\nabla \cdot \boldsymbol u = 0 & \text{ in } \Omega \times (0, T) \\

\boldsymbol u = \boldsymbol g & \text{ on } \Gamma_D \times (0, T) \\

\sigma (\boldsymbol u, p) \boldsymbol{\hat n} = \boldsymbol h & \text{ on } \Gamma_N \times (0, T) \\

\boldsymbol u(0)= \boldsymbol u_0 & \text{ in } \Omega \times \{ 0\}

\end{cases}

being $\backslash boldsymbol\; u$ the fluid velocity, $p$ the fluid pressure, $\backslash boldsymbol\; f$ a given forcing term, $\backslash boldsymbol\{\backslash hat\; n\}$ the outward directed unit normal vector to $\backslash Gamma\_N$, and $\backslash boldsymbol\; \backslash sigma\; (\backslash boldsymbol\; u,\; p)$ the viscous stress tensor defined as:

:$$

\boldsymbol \sigma (\boldsymbol u, p) = -p \boldsymbol I + 2 \mu \boldsymbol \epsilon (\boldsymbol u).

Let $\backslash mu$ be the dynamic viscosity of the fluid, $\backslash boldsymbol\; I$ the second order identity tensor and $\backslash boldsymbol\; \backslash epsilon\; (\backslash boldsymbol\; u)$ the strain-rate tensor defined as:

:$$

\boldsymbol \epsilon (\boldsymbol u) = \frac{1}{2} ((\nabla \boldsymbol u) + (\nabla \boldsymbol u)^T).

The functions $\backslash boldsymbol\; g$ and $\backslash boldsymbol\; h$ are given Dirichlet and Neumann boundary data, while $\backslash boldsymbol\; u\_0$ is the initial condition.

In order to find a variational formulation of the Navier–Stokes equations, consider the following infinite-dimensional spaces:

:$$

\mathcal V_g= \{ \boldsymbol u \in [H^1(\Omega)]^d : \boldsymbol u = \boldsymbol g \text{ on } \Gamma_D \},

:$$

\mathcal V_0 = [H^1_0(\Omega)]^d=\{ \boldsymbol u \in [H^1(\Omega)]^d : \boldsymbol u = \boldsymbol 0 \text{ on } \Gamma_D \},

:$$

\mathcal Q =L^2(\Omega).

Furthermore, let $\backslash boldsymbol\; \backslash mathcal\; V\_g\; =\; \backslash mathcal\; V\_g\; \backslash times\; \backslash mathcal\; Q$ and $\backslash boldsymbol\; \backslash mathcal\; V\_0\; =\; \backslash mathcal\; V\_0\; \backslash times\; \backslash mathcal\; Q$. The weak form of the unsteady-incompressible Navier–Stokes equations reads: given $\backslash boldsymbol\; u\_\{0\}$,

:$$

\forall t \in (0, T), \; \text{find } (\boldsymbol u, p) \in \boldsymbol \mathcal V_g \text{ such that }

:$$

\begin{align}

\bigg( \boldsymbol v, \rho \dfrac{\partial \boldsymbol u}{\partial t}\bigg) + a (\boldsymbol v, \boldsymbol u) + c (\boldsymbol v, \boldsymbol u, \boldsymbol u) - b (\boldsymbol v, p) + b (\boldsymbol u, q) = (\boldsymbol v, \boldsymbol f) + (\boldsymbol v, \boldsymbol h)_{\Gamma_N} \; \; \forall (\boldsymbol v, q) \in \boldsymbol \mathcal V_0

\end{align}

where $(\backslash cdot,\; \backslash cdot)$ represents the $L^2(\backslash Omega)$ inner product and $(\backslash cdot,\; \backslash cdot)\_\{\backslash Gamma\_N\}$ the $L^2(\backslash Gamma\_N)$ inner product. Moreover, the bilinear forms $a(\backslash cdot,\; \backslash cdot)$, $b(\backslash cdot,\; \backslash cdot)$ and the trilinear form $c(\backslash cdot,\; \backslash cdot,\; \backslash cdot)$ are defined as follows:

:$$

\begin{align}

a (\boldsymbol v, \boldsymbol u) = & (\nabla \boldsymbol v, \mu ((\nabla \boldsymbol u) + (\nabla \boldsymbol u)^T)), \\

b (\boldsymbol v, q) = &(\nabla \cdot \boldsymbol v, q), \\

c (\boldsymbol v, \boldsymbol u, \boldsymbol u) = &(\boldsymbol v, \rho (\boldsymbol u \cdot \nabla) \boldsymbol u).

\end{align}

In order to discretize in space the Navier–Stokes equations, consider the function space of finite element

:$$

X_r^h = \{u^h \in C^0 (\overline\Omega): u^h|_k \in \mathbb P_r, \; \forall k \in \Tau_h\}

of piecewise Lagrangian Polynomials of degree $r\; \backslash geq\; 1$ over the domain $\backslash Omega$ triangulated with a mesh $\backslash Tau\_h$ made of tetrahedrons of diameters $h\_k$, $\backslash forall\; k\; \backslash in\; \backslash Tau\_h$.

Following the approach shown above, let introduce a multiscale direct-sum decomposition of the space $\backslash boldsymbol\; \backslash mathcal\; V$ which represents either $\backslash boldsymbol\; \backslash mathcal\; V\_g$ and $\backslash boldsymbol\; \backslash mathcal\; V\_0$:{{cite journal |last1=Forti |first1=Davide |last2=Dedè |first2=Luca |title=Semi-implicit BDF time discretization of the Navier–Stokes equations with VMS-LES modeling in a High Performance Computing framework |journal=Computers & Fluids |date=August 2015 |volume=117 |pages=168–182 |doi=10.1016/j.compfluid.2015.05.011}}

:$$

\boldsymbol \mathcal V = \boldsymbol \mathcal V_h \oplus \boldsymbol \mathcal V',

being

:$$

\boldsymbol \mathcal V_h = \mathcal V_{g_h} \times \mathcal Q \text{ or } \boldsymbol \mathcal V_h = \mathcal V_{0_h} \times \mathcal Q

the finite dimensional function space associated to the coarse scale, and

:$$

\boldsymbol \mathcal V' = \mathcal V_g' \times \mathcal Q \text{ or } \boldsymbol \mathcal V' = \mathcal V_0' \times \mathcal Q

the infinite-dimensional fine scale function space, with

:$\backslash mathcal\; V\_\{g\_h\}\; =\; \backslash mathcal\; V\_g\; \backslash cap\; X\_r^h$,

:$\backslash mathcal\; V\_\{0\_h\}\; =\; \backslash mathcal\; V\_0\; \backslash cap\; X\_r^h$

and

:$\backslash mathcal\; Q\_h\; =\; \backslash mathcal\; Q\; \backslash cap\; X\_r^h$.

An overlapping sum decomposition is then defined as:

:$$

\begin{align}

& \boldsymbol u = \boldsymbol u^h + \boldsymbol u' \text{ and } p = p^h + p' \\

& \boldsymbol v = \boldsymbol v^h + \boldsymbol v' \;\text{ and } q = q^h + q'

\end{align}

By using the decomposition above in the variational form of the Navier–Stokes equations, one gets a coarse and a fine scale equation; the fine scale terms appearing in the coarse scale equation are integrated by parts and the fine scale variables are modeled as:

:$$

\begin{align}

\boldsymbol u' \approx & -\tau_M (\boldsymbol u^h) \boldsymbol r_M (\boldsymbol u^h, p^h), \\

p' \approx & -\tau_C (\boldsymbol u^h) \boldsymbol r_C (\boldsymbol u^h).

\end{align}

In the expressions above, $\backslash boldsymbol\; r\_M\; (\backslash boldsymbol\; u^h,\; p^h)$ and $\backslash boldsymbol\; r\_C\; (\backslash boldsymbol\; u^h)$ are the residuals of the momentum equation and continuity equation in strong forms defined as:

:$$

\begin{align}

\boldsymbol r_M (\boldsymbol u^h, p^h) = & \rho \dfrac{\partial \boldsymbol u^h}{\partial t} + \rho (\boldsymbol u^h \cdot \nabla) \boldsymbol u^h - \nabla \cdot \boldsymbol \sigma (\boldsymbol u^h, p^h) - \boldsymbol f,\\

\boldsymbol r_C (\boldsymbol u^h) = & \nabla \cdot \boldsymbol u^h,

\end{align}

while the stabilization parameters are set equal to:

:$$

\begin{align}

\tau_M (\boldsymbol u^h) = & \bigg ( \frac{\sigma^2 \rho^2}{\Delta t^2} + \frac{\rho^2 }{h_k^2} |\boldsymbol u^h|^2 + \frac{\mu^2}{h_k^4}C_r\bigg )^{-1/2}, \\

\tau_C (\boldsymbol u^h) = & \frac{h_k^2}{\tau_M (\boldsymbol u^h) },

\end{align}

where $C\_r\; =\; 60\; \backslash cdot\; 2^\{r-2\}$ is a constant depending on the polynomials's degree $r$, $\backslash sigma$ is a constant equal to the order of the backward differentiation formula (BDF) adopted as temporal integration scheme and $\backslash Delta\; t$ is the time step. The semi-discrete variational multiscale multiscale formulation (VMS-LES) of the incompressible Navier–Stokes equations, reads: given $\backslash boldsymbol\; u\_\{0\}$,

:$$

\forall t \in (0, T), \; \text{find } \boldsymbol U^h = \{\boldsymbol u^h, p^h\} \in \boldsymbol \mathcal V_{g_h} \text{ such that } A(\boldsymbol V^h, \boldsymbol U^h ) = F(\boldsymbol V^h) \; \; \forall \boldsymbol V^h= \{\boldsymbol v^h, q^h\} \in \boldsymbol \mathcal V_{0_h},

being

:$$

A(\boldsymbol V^h, \boldsymbol U^h ) = A^{NS}(\boldsymbol V^h, \boldsymbol U^h ) + A^{VMS}(\boldsymbol V^h, \boldsymbol U^h ),

and

:$$

F(\boldsymbol V^h) = (\boldsymbol v, \boldsymbol f) + (\boldsymbol v, \boldsymbol h)_{\Gamma_N}.

The forms $A^\{NS\}(\backslash cdot,\; \backslash cdot)$ and $A^\{VMS\}(\backslash cdot,\; \backslash cdot)$ are defined as:

:$$

\begin{align}

A^{NS}(\boldsymbol V^h, \boldsymbol U^h )= & \bigg( \boldsymbol v^h, \rho \dfrac{\partial \boldsymbol u^h}{\partial t}\bigg) + a (\boldsymbol v^h, \boldsymbol u^h) + c (\boldsymbol v^h, \boldsymbol u^h, \boldsymbol u^h) - b (\boldsymbol v^h, p^h) + b (\boldsymbol u^h, q^h), \\

A^{VMS}(\boldsymbol V^h, \boldsymbol U^h ) = & \underbrace{\big( \rho \boldsymbol u^h \cdot \nabla \boldsymbol v^h + \nabla q^h, \tau_M(\boldsymbol u^h) \boldsymbol r_M(\boldsymbol u^h, p^h) \big)}_{\text{SUPG}} - \underbrace{(\nabla \cdot \boldsymbol v^h, \tau_c(\boldsymbol u_h)\boldsymbol r_C(\boldsymbol u^h)) + \big( \rho \boldsymbol u^h \cdot (\nabla \boldsymbol u^h)^T, \tau_M(\boldsymbol u^h) \boldsymbol r_M (\boldsymbol u^h, p^h) \big )}_{\text{VMS}} - \underbrace{(\nabla \boldsymbol v^h, \tau_M(\boldsymbol u^h) \boldsymbol r_M(\boldsymbol u^h, p^h) \otimes \tau_M(\boldsymbol u^h) \boldsymbol r_M(\boldsymbol u^h, p^h) )}_{\text{LES}}.

\end{align}

From the expressions above, one can see that:

• the form $A^\{NS\}(\backslash cdot,\; \backslash cdot)$ contains the standard terms of the Navier–Stokes equations in variational formulation;
• the form $A^\{VMS\}(\backslash cdot,\; \backslash cdot)$ contain four terms:

1. the first term is the classical SUPG stabilization term;
2. the second term represents a stabilization term additional to the SUPG one;
3. the third term is a stabilization term typical of the VMS modeling;
4. the fourth term is peculiar of the LES modeling, describing the Reynolds cross-stress.

• Navier–Stokes equations
• Large eddy simulation
• Finite element method
• Backward differentiation formula
• Computational fluid dynamics
• Streamline upwind Petrov–Galerkin pressure-stabilizing Petrov–Galerkin formulation for incompressible Navier–Stokes equations

Category:Mathematical modeling

Category:Numerical analysis

Category:Computational fluid dynamics