변동 다중점법

가변 멀티스케일 방식(VMS)은 멀티스케일 현상을 위한 모델과 숫자 방법을 도출하는 데 사용되는 기법이다.^[1]VMS 프레임워크는 단수 섭동과 유한 요소 공간과의 호환성 조건 모두에서 표준 갤러킨 방법의 안정성이 보장되지 않는 설계 안정화된 유한 요소 방법에 주로 적용되어 왔다.^[2]

안정화된 방법은 표준 Galerkin 방법의 대표적인 단점인 부착 중심 흐름 문제 및 보간 기능의 임의 조합이 불안정한 형성에 귀속될 수 있는 문제를 해결하도록 설계되었기 때문에 계산 유체 역학에서 관심이 높아지고 있다.^[3][4]이 등급의 문제들에 대한 안정화된 방법의 이정표는 압축 불가능한 Navier의 대류 지배 흐름을 위해 80년대에 설계된 SUPG(Simply Upwind Petrov-Galerkin method)로 간주될 수 있다.-브룩스와 ^[5][6]휴즈의 스톡스 방정식VMS(Variative Multiscale Method)는 1995년 휴즈에 의해 도입되었다.^[7]광범위하게 말하면, VMS는 멀티스케일 현상을 포착할 수 있는 수학적 모델과 수치적 방법을 얻기 위해 사용되는 기법이다.^[1] 사실, VMS는 대개 다수의 척도 그룹으로 구분되는 큰 범위의 문제에 채택된다.^[8]이 방법의 주요 아이디어는 용액의 총분해를 $u{\displaystyle }$ =${\displaystyle u\stackrel{}{}}$=${\displaystyle }$u = ${\displaystyle {}^{}}$ ${\displaystyle {\prime }^{}}$ ${\$$bar}+u}$로 설계하는 것인데$,$ $u{\displaystyle \stackrel{}{}}$= ${\displaystyle$ {\u}}}은(는) 거친 규모의 솔루션으로 표시되고$$ 숫자적으로 해결되는 반면 $u{\displaystyle {}^{}}$ u${\displaystyle {}^{}}$′ ${\$은 미세한 솔루션을 나타낸다$$.거친 척도 방정식의 문제에서 그것을 분석적으로 제거한다.^[1]

추상적인 틀

변동형식의 추상적 디리클레 문제

매끄러운 경계$$ $\gamma {\displaystyle \mathrm{}}$⊂ ⊂ R d${\displaystyle }$⊂ ${\displaystyle \mathrm{r}}$ r ${\displaystyle {}^{}}$r \$Displaystyle \Oomega \subset \$${\displaystyle {\mathbb{}}^{}}$ $}{\$ d ${\displaystyle \ge }$ $1{\displaystyle }$ ${\daystylevelopeq}$을 공간 치수로 간주한다.$일반$$,$ 두 번째 순서, $비대칭$ 차동 연산자 L {\displaystyle {\$mathcal {L}$을(를) 사용하여 나타내는 다음과 같은 경계 값 문제를 고려하십시오.^[4]

${\displaystyle {\text{find}}u:\Oomega \to \mathb{R}{\text{{{{{{{}}}:}$

${\displaystyle {\begin{case}{\mathcal{L}u=f&{\text{{}}}\Oomega \u=g&{\text{{}}}}}}$

$being{\displaystyle }$ f :${\displaystyle \mathrm{\Omega }}$→ ${\displaystyle R\mathbb{}}$ ${\$$\Oomega \to \mathb {R}$및 $g$ : ${\displaystyle \gamma \mathbb{}}$→ R ${\$이(가) 주어진$$ 함수.H ${\displaystyle {1\mathrm{}}^{}}$(${\displaystyle \mathrm{\Omega }}$) ${\displaystyle$ H$^{1}(\Oomega )}$을(를) 제곱 통합형 파생상품이 있는 사각형 통합 함수의 힐버트 공간으로 두십시오$$.^[4]

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

시험용액$$ 공간 V ${\displaystyle g_{}}$ ${\$과(와) 다음과 같이$$정의된 가중 $함수$ 공간 V {\$displaystyle {\mathcal$ {V}을(를$)$ 고려하십시오.^[4]

${\displaystyle {\mathcal {V}_{g}=\{u\in H^{1}(\Oomega ):\,u=g{\text{{}}}}}$

${\displaystyle {\mathcal{V}}=H_{0}^{1}^{1}(\Oomega )=\{v\in H^{1}(\Oomega ):\,v=0{\text{{}}{}}}$

위에서 정의한 경계 값 문제의 변동 공식은 다음과 같다.^[4]

${\displaystyle \text{다음}}$과 ${\displaystyle \mathrm{같은}}$ g V ${\displaystyle g_{}}$를 찾으십시오${\displaystyle .a}$ (${\displaystyle v}$ ,${\displaystyle u}$ ${\displaystyle )}$ = ${\displaystyle f}$ ( v${\displaystyle }$)${\displaystyle \mathrm{}v\mathcal{}}$v {$$v {\$displaystyle${\$text}}{}u\in {\mathcal {V}_{g}{\text}{{}}}}, },$ }$a(v,u)=f(v)\,\,\,\,\,\,\,},\,},}},\,},},\,\,},},\,},},\c$

being ${\displaystyle a(v,u)}$ the bilinear form satisfying ${\displaystyle a(v,u)=(v,{\mathcal {L}}u)}$, ${\displaystyle f(v)=(v,f)}$ a bounded linear functional on ${\displaystyle {\mathcal {V}}}$ and ${\displaystyle (\cdot ,\cdot )}$ is the L ${\displaystyle {2\mathrm{}}^{}}$(${\displaystyle \mathrm{\Omega }}$) ${\displaystyle L^{2}(\Oomega )}$ 내부 제품$$.^[2]Furthermore, the dual operator ${\displaystyle {\mathcal {L}}^{*}}$ of ${\displaystyle {\mathcal {L}}}$ is defined as that differential operator such that ${\displaystyle {\mathcal {(}}v,{\mathcal {L}}u)=({\mathcal {L}}^{*}v,u)\,\,\,\forall u,\,v\in$${\mathcal{V$^[7]

VMS 접근법에서 기능 공간은 V ${\displaystyle g_{}}$ ${\$discale {\$displaystyle${\$mathcal$ {$V}$$\_$${g}$ 및 V ${\displaystyle$ {\$mathcal {V}$ 모두에 대해 멀티스케일 직접 합계 분해를 통해 다음과$$ 같이 분해된다.^[1]

$mathcal {V}={\bar {\mathcal {V}\plus {\mathcal}}}}}}}}}:{\mathcal {V}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}?$

그리고

${\displaystyle {\mathcal {V}_{g}={\bar {{\mathcal {V}}\plus {\mathcal {V}_{g}'''''''''이다.}$

$따라서{\displaystyle }$ $u{\displaystyle }$ ${\displaystyle u}$ 및$$ v ${\displaystyle v}$에 대해 다음과 같이$$ 중복된 합계 분해를 가정한다.

${\displaystyle u}$= ${\displaystyle u\stackrel{\text{'}}{}}$+ ${\displaystyle u}$ ${\displaystyle {\prime }^{}}$ 및 v${\displaystyle }$= ${\displaystyle v\stackrel{}{}}$′ + ${\displaystyle {}^{}{\prime }^{}}$ ${\$

where ${\displaystyle {\bar {u}}}$ represents the coarse (resolvable) scales and ${\displaystyle u'}$ the fine (subgrid) scales, with ${\displaystyle {\bar {u}}\in {\bar {{\mathcal {V}}_{g}}}}$, ${\displaystyle {u'}\in {{\mathcal {V}}_{g}}'}$, ${\displaystyle }$${\displaystyle \stackrel{}{V}\in }$ $'{\$ {$V}$ 및 $v$ ′${\displaystyle {}^{}\in }$ ${\displaystyle {\mathcal{}}^{}}$ ${\displaystyle {\prime }^{}}$ ${\$ v$'\in {\mathcal$ {$V$ 특히 이러한 기능에 대해서는 다음과 같은 가정을 한다.^[1]

${\displaystyle{\begin{정렬}{\bar{너}}=g&,&{\text{에}}\Gamma&&\forall&{\bar{너}}\in{\bar{\mathcal{V_{g}}}},\\u'=0&,&{\text{에}}\Gamma&&\forall&u'\in{\mathcal{V_{g}}}',\\{\bar{v}}=0&,&{\text{에}}\Gamma&&\forall&{\bar{v}}\in{\bar{{V\mathcal}}},\\v'=0&, &,{\text{에.}}\Gamma&&\forall&v'\in{{V\mathcal}}'.\end{정렬}}}$

이것을 염두에 두고, 변량형식을 다음과 같이 다시 쓸 수 있다.

${\displaystyle a({\bar {v}+v',{\bar {u}+u'=ff\bar {v}+v')}$

$그리고$, $a{\displaystyle }$ (${\displaystyle \cdot }$ ,${\displaystyle }$$⋅$${\displaystyle }$)의$이선성$과 $f{\displaystyle }$ ($$ )의 선형성을$$ 사용함으로써$,$

${\displaystyle a({\bar{v},{\bar {u},{v},u')+a(v'),{\bar {u}+a(v',u')=fbar\bar {v}+f(v')}$

마지막 방정식, 거친 척도 및 미세 척도 문제:

${\displaystyle{\text{}}}{\bar {u}\in {\bar {\mathcal {V}_{g}{\u'in {\mathcal {V}{\text}}}}}in {\mathcal}{\text{}}in}}}}}in {}}}}}}}}}}}}}}}}}}}}in {}$

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

또는 동등하게 a ${\displaystyle (v}$, ${\displaystyle u}$ ${\displaystyle )}$=${\displaystyle }$( v${\displaystyle }$, ${\displaystyle L\mathcal{}}$ ${\displaystyle u}$) ${\displaystyle a(v,u)=(v,{\mathcal{L}u)}$ 및$$ $f{\displaystyle (v}$ )${\displaystyle }$=${\displaystyle }$(${\displaystyle v}$, f${\displaystyle }$) = ${\displaystystyle f(v$,f$)=(v,$f$)\}\}:$

${\displaystyle{\text{}}}{\bar {u}\in {\bar {\mathcal {V}_{g}{\u'in {\mathcal {V}{\text}}}}}in {\mathcal}{\text{}}in}}}}}in {}}}}}}}}}}}}}}}}}}}}in {}$

${\displaystyle {\begin{aligned}&&({\bar {v}},{\mathcal {L}}{\bar {u}})+({\bar {v}},{\mathcal {L}}u')&=({\bar {v}},f)&&\forall {\bar {v}}\in {\bar {\mathcal {V}}},\\&&(v',{\mathcal {L}}{\bar {u}})+(v',{\mathcal {L}}u')&=(v',f)&&\forall v'\in {\mathcal {V}}'.\\end{aigned}}$

$({\displaystyle {}^{}}$ ${\displaystyle {\prime }^{}}$, ${\displaystyle L\mathcal{}}$ ${\displaystyle u{}^{}}$ ${\displaystyle {\prime }^{}}$)${\displaystyle }$=${\displaystyle }$-${\displaystyle }$( ${\displaystyle {}^{}}$ ${\displaystyle {\prime }^{}}$, ${\displaystyle L\mathcal{}}$ ${\displaystyle u\stackrel{}{}\text{'}}$s -${\displaystyle f}$ $){\displaystyle (v',{\mathcal{L}u$$v'),{\mathcal{L}{\bar{u$}-$f)$로 두 번째 문제를 재배열하여 해당하는 오일러-라그랑 방정식은 다음과 같다.^[7]

${\displaystyle {\begin{case}{\mathcal {{L}{{\mathcal{{L}}{\bar{u}-f)&{\text{{}}}}}}}}$

미세 스케일 솔루션 $′{\$$displaystyle$$u'}$은(는$)$ 거친 스케일 방정식 $u$의 강한 잔차에 따라$$ 결정된다는 것을 보여준다$.$^[7]The fine scale solution can be expressed in terms of ${\displaystyle {\mathcal {L}}{\bar {u}}-f}$ through the Green's function ${\displaystyle G:\Omega \times \Omega \to \mathbb {R} {\text{ with }}G=0{\text{ on }}\Gamma \times \Gamma }$:

${\displaystyle u'=-\int _{\Oomega}G(x,y)({\mathcal{{\\bar{u}-f)(x)\,d\Oomega _{x}\,\,\forall y\in \Oomega.}$

$\Delta {\displaystyle }$ ${\displaystyle \delta }$을(를) Dirac 델타 함수로$$ 두십시오. 정의상${\displaystyle \mathrm{}\mathrm{by}}$ y${\displaystyle \mathrm{\Omega }}$ ${\displaystyle \forall y\in \Oomega}$을(를) 해결하면 Green의 함수를 찾을 수 있다.

${\displaystyle {\begin{case}{\mathcal {L}^{*G(x,y)=\delta(x-y)&{\text{{}}}}}\Oomega \\G(x,y)=0&{}}}}}}}}\Mathcal {L}}$

또한$$, 차동 연산자와 ${\displaystyle {}^{}}$한${\displaystyle {}^{\mathrm{새로운}}}$차동$$$연산자$ M {\$displaystyle {\mathcal{M}}$의 $관점$에서 $u${\$displaystyle$$-{\mathcal{L}^{-1}$을 표현하면 된다$.{\displaystyle }$

${\displaystyle u'={\mathcal {M}}({\mathcal {L}}{\bar {u}}-f),}$ with ${\displaystyle {\mathcal {M}}\approx -{\mathcal {L}}^{-1}}$. In order to eliminate the explicit dependence in the coarse scale equation of the sub-grid scale terms, considering the definition of the dual연산자, 마지막 식은 굵은 척도 방정식의 두 번째 항에서 대체할 수 있다.^[1]

${\displaystyle({\bar {{v},{\mathcal {L}^{*}{\bar{v},u')=({\mathcal {L}^{{v},{\mathcal {{L}}},{\mathcal {{}}-f).}$

Since ${\displaystyle {\mathcal {M}}}$ is an approximation of ${\displaystyle -{\mathcal {L}}^{-1}}$, the Variational Multiscale Formulation will consist in finding an approximate solution ${\displaystyle {\tilde {\bar {u}}}\approx {\bar {u}}}$ instead of ${\displays$$tyle {\bar {u$그러므로 거친 문제는 다음과 같이 다시 쓰여진다.^[1]

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

존재

${\displaystyle ({\mathcal {L}}^{*}{\bar {v}},{\mathcal {M}}({\mathcal {L}}{\tilde {\bar {u}}}-f))=-\int _{\Omega }\int _{\Omega }({\mathcal {L}}^{*}{\bar {v}})(y)G(x,y)({\mathcal {L}}{\tilde {\bar {u}}}-f)(x)\,d\Omega _{x}\,d\Omega _{y}.}$

양식 소개

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

그리고 기능적

${\displaystyle L(v\stackrel{}{}¯,G}$)${\displaystyle }$= ( ${\displaystyle v\stackrel{}{}}$∗, f${\displaystyle }$)${\displaystyle }$+${\displaystyle }$( ${\displaystyle {\mathcal{}}^{}}$ ${\displaystyle {\ast }^{}}$ v ${\displaystyle {}^{}\stackrel{}{},}$ M $){\displaystyle L({\bar {{v},G)=({\bar {{v},f)+({\mathcal{L}^{v},{v},{},{mathcal {{M}f$

굵은 스케일 방정식의 VMS 제형은 다음과 같이 재배열된다.^[7]

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

일반적으로 $M{\displaystyle \mathcal{}}$ ${\$과(와) $G{\displaystyle }$ ${\displaystyle$ G$}$을(를) 모두 결정할 수 없으므로$,$ 일반적으로 근사치를 채택한다.이러한 의미에서 거친 스케일 공간 ${\$ {V$}_{g}}$ 및 $V$ ${\${\$mathcal$ {V$}}$은(는) 다음과 같이 함수의 유한 치수 공간으로 선택된다$$.^[1]

${\displaystyle {\bar}{V}_{g}}\equiv {\mathcal{V}_{{h}}}}:={\mathcal {V}_{g}\cap X_{r}^{h}(\Oomega )}$

그리고

${\displaystyle {\bar {\mathcal {V}}\equiv {\mathcal {V}_{h}:={\mathcal{V}\cap X_{h}^{r}(\Oomega ),}$

되는 것은 Xrh(Ω){\displaystyle X_{r}(\Omega)}유한 요소 공간의 라그랑주 다항식의 정도 r≥ 1{\displaystyle r\geq 1}에 대한 그물코에Ω{\displaystyle \Omega}.[4] 있다는 점을 Vg({\displaystyle{{V\mathcal}}'}. 이다무한 차원 공간인 반면, $V{\displaystyle {\mathcal{}}_{}}$ ${\displaystyle g_{{}_{}}}$ ${\displaystyle {h_{}}_{}}$ ${\${\$mathcal {V}_{g_{h},$ $V{\displaystyle {\mathcal{}}_{}}$ ${\displaystyle h_{}}$ ${\${\$mathcal {V}_{h}$는 유한 차원 공간이다$$.

Let ${\displaystyle u_{h}\in {\mathcal {V}}_{g_{h}}}$ and ${\displaystyle v_{h}\in {\mathcal {V}}_{h}}$ be respectively approximations of ${\displaystyle {\tilde {\bar {u}}}}$ and ${\displaystyle {\bar {v}}}$, and let ${\displaystyle {$$\tilde{G}}$ 및$$ $M{\displaystyle \stackrel{}{\mathcal{}}}$ ~ ${\$mathcal{$M}}은($는) $각각$ $G{\displaystyle }$ ${\displaystyle$ G$}$및 M ${\$의 근사치여야$$ 한다$.$유한 요소 근사치의 VMS 문제는 다음과 같다.^[7]

${\displaystyle {\text{find }}u_{h}\in {\mathcal {V}}_{g_{h}}:B(v_{h},u_{h},{\tilde {G}})=L(v_{h},{\tilde {G}})\,\,\,\forall {v}_{h}\in {\mathcal {V}}_{h}}$

or, equivalently:

${\displaystyle {\text{find }}u_{h}\in {\mathcal {V}}_{g_{h}}:a(v_{h},u_{h})+({\mathcal {L}}^{*}v_{h},{\mathcal {\tilde {M}}}({\mathcal {L}}{u_{h}}-f))=(v_{h},f)\,\,\,\forall {v}_{h}\in {\mathcal {V}}_{h}}$

VMS and stabilized methods

Consider an advection–diffusion problem:^[4]

${\displaystyle {\begin{cases}-\mu \Delta u+{\boldsymbol {b}}\cdot \nabla u=f&{\text{ in }}\Omega \\u=0&{\text{ on }}\partial \Omega \end{cases}}}$

where ${\displaystyle \mu \in \mathbb {R} }$ is the diffusion coefficient with ${\displaystyle \mu >0}$ and ${\displaystyle {\boldsymbol {b}}\in \mathbb {R} ^{d}}$ is a given advection field. Let ${\displaystyle {\mathcal {V}}=H_{0}^{1}(\Omega )}$ and ${\displaystyle u\in {\mathcal {V}}}$, ${\displaystyle {\boldsymbol {b}}\in [L^{2}(\Omega )]^{d}}$, ${\displaystyle f\in L^{2}(\Omega )}$.^[4] Let ${\displaystyle {\mathcal {L}}={\mathcal {L}}_{diff}+{\mathcal {L}}_{adv}}$, being ${\displaystyle {\mathcal {L}}_{diff}=-\mu \Delta }$ and ${\displaystyle {\mathcal {L}}_{adv}={\boldsymbol {b}}\cdot \nabla }$.^[1] The variational form of the problem above reads:^[4]

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

being

${\displaystyle a(v,u)=(\nabla v,\mu \nabla u)+(v,{\boldsymbol {b}}\cdot \nabla u).}$

Consider a Finite Element approximation in space of the problem above by introducing the space ${\displaystyle {\mathcal {V}}_{h}={\mathcal {V}}\cap X_{h}^{r}}$ over a grid ${\displaystyle \Omega _{h}=\bigcup _{k=1}^{N}\Omega _{k}}$ made of ${\displaystyle N}$ elements, with ${\displaystyle u_{h}\in {\mathcal {V}}_{h}}$.

The standard Galerkin formulation of this problem reads^[4]

${\displaystyle {\text{find }}u_{h}\in {\mathcal {V}}_{h}:\;\;\;a(v_{h},u_{h})=(f,v_{h})\;\;\;\forall v\in {\mathcal {V}},}$

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

${\displaystyle {\text{ find }}u_{h}\in {\mathcal {V}}_{h}:\,\,\,a(v_{h},u_{h})+{\mathcal {L}}_{h}(u_{h},f;v_{h})=(f,v_{h})\,\,\,\forall v_{h}\in {\mathcal {V}}_{h}}$

for a suitable form ${\displaystyle {\mathcal {L}}_{h}}$ that satisfies:^[4]

${\displaystyle {\mathcal {L}}_{h}(u,f;v_{h})=0\,\,\,\forall v_{h}\in {\mathcal {V}}_{h}.}$

The form ${\displaystyle {\mathcal {L}}_{h}}$ can be expressed as ${\displaystyle (\mathbb {L} v_{h},\tau ({\mathcal {L}}u_{h}-f))_{\Omega _{h}}}$, being ${\displaystyle \mathbb {L} }$ a differential operator such as:^[1]

${\displaystyle \mathbb {L} ={\begin{cases}+{\mathcal {L}}&\,\,\,&{\text{ Galerkin/least squares (GLS)}}\\+{\mathcal {L}}_{adv}&\,\,\,&{\text{ Streamline Upwind Petrov-Galerkin (SUPG)}}\\-{\mathcal {L}}^{*}&\,\,\,&{\text{ Multiscale}}\\\end{cases}}}$

and ${\displaystyle \tau }$ is the stabilization parameter. A stabilized method with ${\displaystyle \mathbb {L} =-{\mathcal {L}}^{*}}$ 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

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

or, in terms of the Green's function as

${\displaystyle \tau \delta (x-y)={\tilde {G}}(x,y)\approx G(x,y),}$

which yields the following definition of ${\displaystyle \tau }$:

${\displaystyle \tau ={\frac {1}{ \Omega _{k} }}\int _{\Omega _{k}}\int _{\Omega _{k}}G(x,y)\,d\Omega _{x}\,d\Omega _{y}.}$^[7]

VMS turbulence modeling for large-eddy simulations of incompressible flows

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.^[9][10]

Incompressible Navier–Stokes equations

Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density ${\displaystyle \rho }$ in a domain ${\displaystyle \Omega \in \mathbb {R} ^{d}}$ with boundary ${\displaystyle \partial \Omega =\Gamma _{D}\cup \Gamma _{N}}$, being ${\displaystyle \Gamma _{D}}$ and ${\displaystyle \Gamma _{N}}$ portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied (${\displaystyle \Gamma _{D}\cap \Gamma _{N}=\emptyset }$):^[4]

${\displaystyle {\begin{cases}\rho {\dfrac {\partial {\boldsymbol {u}}}{\partial t}}+\rho ({\boldsymbol {u}}\cdot \nabla ){\boldsymbol {u}}-\nabla \cdot {\boldsymbol {\sigma }}({\boldsymbol {u}},p)={\boldsymbol {f}}&{\text{ in }}\Omega \times (0,T)\\\nabla \cdot {\boldsymbol {u}}=0&{\text{ in }}\Omega \times (0,T)\\{\boldsymbol {u}}={\boldsymbol {g}}&{\text{ on }}\Gamma _{D}\times (0,T)\\\sigma ({\boldsymbol {u}},p){\boldsymbol {\hat {n}}}={\boldsymbol {h}}&{\text{ on }}\Gamma _{N}\times (0,T)\\{\boldsymbol {u}}(0)={\boldsymbol {u}}_{0}&{\text{ in }}\Omega \times \{0\}\end{cases}}}$

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

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

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

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

The functions ${\displaystyle {\boldsymbol {g}}}$ and ${\displaystyle {\boldsymbol {h}}}$ are given Dirichlet and Neumann boundary data, while ${\displaystyle {\boldsymbol {u}}_{0}}$ is the initial condition.^[4]

Global space time variational formulation

In order to find a variational formulation of the Navier–Stokes equations, consider the following infinite-dimensional spaces:^[4]

${\displaystyle {\mathcal {V}}_{g}=\{{\boldsymbol {u}}\in [H^{1}(\Omega )]^{d}:{\boldsymbol {u}}={\boldsymbol {g}}{\text{ on }}\Gamma _{D}\},}$

${\displaystyle {\mathcal {V}}_{0}=[H_{0}^{1}(\Omega )]^{d}=\{{\boldsymbol {u}}\in [H^{1}(\Omega )]^{d}:{\boldsymbol {u}}={\boldsymbol {0}}{\text{ on }}\Gamma _{D}\},}$

${\displaystyle {\mathcal {Q}}=L^{2}(\Omega ).}$

Furthermore, let ${\displaystyle {\boldsymbol {\mathcal {V}}}_{g}={\mathcal {V}}_{g}\times {\mathcal {Q}}}$ and ${\displaystyle {\boldsymbol {\mathcal {V}}}_{0}={\mathcal {V}}_{0}\times {\mathcal {Q}}}$. The weak form of the unsteady-incompressible Navier–Stokes equations reads:^[4] given ${\displaystyle {\boldsymbol {u}}_{0}}$,

${\displaystyle \forall t\in (0,T),\;{\text{find }}({\boldsymbol {u}},p)\in {\boldsymbol {\mathcal {V}}}_{g}{\text{ such that }}}$

${\displaystyle {\begin{aligned}{\bigg (}{\boldsymbol {v}},\rho {\dfrac {\partial {\boldsymbol {u}}}{\partial t}}{\bigg )}+a({\boldsymbol {v}},{\boldsymbol {u}})+c({\boldsymbol {v}},{\boldsymbol {u}},{\boldsymbol {u}})-b({\boldsymbol {v}},p)+b({\boldsymbol {u}},q)=({\boldsymbol {v}},{\boldsymbol {f}})+({\boldsymbol {v}},{\boldsymbol {h}})_{\Gamma _{N}}\;\;\forall ({\boldsymbol {v}},q)\in {\boldsymbol {\mathcal {V}}}_{0}\end{aligned}}}$

where ${\displaystyle (\cdot ,\cdot )}$ represents the ${\displaystyle L^{2}(\Omega )}$ inner product and ${\displaystyle (\cdot ,\cdot )_{\Gamma _{N}}}$ the ${\displaystyle L^{2}(\Gamma _{N})}$ inner product. Moreover, the bilinear forms ${\displaystyle a(\cdot ,\cdot )}$, ${\displaystyle b(\cdot ,\cdot )}$ and the trilinear form ${\displaystyle c(\cdot ,\cdot ,\cdot )}$ are defined as follows:^[4]

${\displaystyle {\begin{aligned}a({\boldsymbol {v}},{\boldsymbol {u}})=&(\nabla {\boldsymbol {v}},\mu ((\nabla {\boldsymbol {u}})+(\nabla {\boldsymbol {u}})^{T})),\\b({\boldsymbol {v}},q)=&(\nabla \cdot {\boldsymbol {v}},q),\\c({\boldsymbol {v}},{\boldsymbol {u}},{\boldsymbol {u}})=&({\boldsymbol {v}},\rho ({\boldsymbol {u}}\cdot \nabla ){\boldsymbol {u}}).\end{aligned}}}$

Finite element method for space discretization and VMS-LES modeling

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

${\displaystyle X_{r}^{h}=\{u^{h}\in C^{0}({\overline {\Omega }}):u^{h} _{k}\in \mathbb {P} _{r},\;\forall k\in \mathrm {T} _{h}\}}$

of piecewise Lagrangian Polynomials of degree ${\displaystyle r\geq 1}$ over the domain ${\displaystyle \Omega }$ triangulated with a mesh ${\displaystyle \mathrm {T} _{h}}$ made of tetrahedrons of diameters ${\displaystyle h_{k}}$, ${\displaystyle \forall k\in \mathrm {T} _{h}}$. Following the approach shown above, let introduce a multiscale direct-sum decomposition of the space ${\displaystyle {\boldsymbol {\mathcal {V}}}}$ which represents either ${\displaystyle {\boldsymbol {\mathcal {V}}}_{g}}$ and ${\displaystyle {\boldsymbol {\mathcal {V}}}_{0}}$:^[11]

${\displaystyle {\boldsymbol {\mathcal {V}}}={\boldsymbol {\mathcal {V}}}_{h}\oplus {\boldsymbol {\mathcal {V}}}',}$

being

${\displaystyle {\boldsymbol {\mathcal {V}}}_{h}={\mathcal {V}}_{g_{h}}\times {\mathcal {Q}}{\text{ or }}{\boldsymbol {\mathcal {V}}}_{h}={\mathcal {V}}_{0_{h}}\times {\mathcal {Q}}}$

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

${\displaystyle {\boldsymbol {\mathcal {V}}}'={\mathcal {V}}_{g}'\times {\mathcal {Q}}{\text{ or }}{\boldsymbol {\mathcal {V}}}'={\mathcal {V}}_{0}'\times {\mathcal {Q}}}$

the infinite-dimensional fine scale function space, with

${\displaystyle {\mathcal {V}}_{g_{h}}={\mathcal {V}}_{g}\cap X_{r}^{h}}$,

${\displaystyle {\mathcal {V}}_{0_{h}}={\mathcal {V}}_{0}\cap X_{r}^{h}}$

and

${\displaystyle {\mathcal {Q}}_{h}={\mathcal {Q}}\cap X_{r}^{h}}$.

An overlapping sum decomposition is then defined as:^[10][11]

${\displaystyle {\begin{aligned}&{\boldsymbol {u}}={\boldsymbol {u}}^{h}+{\boldsymbol {u}}'{\text{ and }}p=p^{h}+p'\\&{\boldsymbol {v}}={\boldsymbol {v}}^{h}+{\boldsymbol {v}}'\;{\text{ and }}q=q^{h}+q'\end{aligned}}}$

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:^[10]

${\displaystyle {\begin{aligned}{\boldsymbol {u}}'\approx &-\tau _{M}({\boldsymbol {u}}^{h}){\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h}),\\p'\approx &-\tau _{C}({\boldsymbol {u}}^{h}){\boldsymbol {r}}_{C}({\boldsymbol {u}}^{h}).\end{aligned}}}$

In the expressions above, ${\displaystyle {\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h})}$ and ${\displaystyle {\boldsymbol {r}}_{C}({\boldsymbol {u}}^{h})}$ are the residuals of the momentum equation and continuity equation in strong forms defined as:

${\displaystyle {\begin{aligned}{\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h})=&\rho {\dfrac {\partial {\boldsymbol {u}}^{h}}{\partial t}}+\rho ({\boldsymbol {u}}^{h}\cdot \nabla ){\boldsymbol {u}}^{h}-\nabla \cdot {\boldsymbol {\sigma }}({\boldsymbol {u}}^{h},p^{h})-{\boldsymbol {f}},\\{\boldsymbol {r}}_{C}({\boldsymbol {u}}^{h})=&\nabla \cdot {\boldsymbol {u}}^{h},\end{aligned}}}$

while the stabilization parameters are set equal to:^[11]

${\displaystyle {\begin{aligned}\tau _{M}({\boldsymbol {u}}^{h})=&{\bigg (}{\frac {\sigma ^{2}\rho ^{2}}{\Delta t^{2}}}+{\frac {\rho ^{2}}{h_{k}^{2}}} {\boldsymbol {u}}^{h} ^{2}+{\frac {\mu ^{2}}{h_{k}^{4}}}C_{r}{\bigg )}^{-1/2},\\\tau _{C}({\boldsymbol {u}}^{h})=&{\frac {h_{k}^{2}}{\tau _{M}({\boldsymbol {u}}^{h})}},\end{aligned}}}$

where ${\displaystyle C_{r}=60\cdot 2^{r-2}}$ is a constant depending on the polynomials's degree ${\displaystyle r}$, ${\displaystyle \sigma }$ is a constant equal to the order of the backward differentiation formula (BDF) adopted as temporal integration scheme and ${\displaystyle \Delta t}$ is the time step.^[11] The semi-discrete variational multiscale multiscale formulation (VMS-LES) of the incompressible Navier–Stokes equations, reads:^[11] given ${\displaystyle {\boldsymbol {u}}_{0}}$,

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

being

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

and

${\displaystyle F({\boldsymbol {V}}^{h})=({\boldsymbol {v}},{\boldsymbol {f}})+({\boldsymbol {v}},{\boldsymbol {h}})_{\Gamma _{N}}.}$

The forms ${\displaystyle A^{NS}(\cdot ,\cdot )}$ and ${\displaystyle A^{VMS}(\cdot ,\cdot )}$ are defined as:^[11]

${\displaystyle {\begin{aligned}A^{NS}({\boldsymbol {V}}^{h},{\boldsymbol {U}}^{h})=&{\bigg (}{\boldsymbol {v}}^{h},\rho {\dfrac {\partial {\boldsymbol {u}}^{h}}{\partial t}}{\bigg )}+a({\boldsymbol {v}}^{h},{\boldsymbol {u}}^{h})+c({\boldsymbol {v}}^{h},{\boldsymbol {u}}^{h},{\boldsymbol {u}}^{h})-b({\boldsymbol {v}}^{h},p^{h})+b({\boldsymbol {u}}^{h},q^{h}),\\A^{VMS}({\boldsymbol {V}}^{h},{\boldsymbol {U}}^{h})=&\underbrace {{\big (}\rho {\boldsymbol {u}}^{h}\cdot \nabla {\boldsymbol {v}}^{h}+\nabla q^{h},\tau _{M}({\boldsymbol {u}}^{h}){\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h}){\big )}} _{\text{SUPG}}-\underbrace {(\nabla \cdot {\boldsymbol {v}}^{h},\tau _{c}({\boldsymbol {u}}_{h}){\boldsymbol {r}}_{C}({\boldsymbol {u}}^{h}))+{\big (}\rho {\boldsymbol {u}}^{h}\cdot (\nabla {\boldsymbol {u}}^{h})^{T},\tau _{M}({\boldsymbol {u}}^{h}){\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h}){\big )}} _{\text{VMS}}-\underbrace {(\nabla {\boldsymbol {v}}^{h},\tau _{M}({\boldsymbol {u}}^{h}){\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h})\otimes \tau _{M}({\boldsymbol {u}}^{h}){\boldsymbol {r}}_{M}({\boldsymbol {u}}^{h},p^{h}))} _{\text{LES}}.\end{aligned}}}$

From the expressions above, one can see that:^[11]

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

See also

• 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

References