RBVMS

$$N(u,p) = \rho \left( \frac{\partial u}{\partial t} + (u \cdot \nabla) u \right) + \nabla p - \nabla \cdot (2 \mu \epsilon(u)) - f$$

$$N(u,p) = \rho \left( \frac{\partial u}{\partial t} + (u \cdot \nabla) u \right) + \nabla p - \nabla \cdot (2 \mu \epsilon(u)) - f$$

\documentclass{article} \usepackage{amsmath} \begin{document} \begin{center} ( \pmb{\mathcal{N}(\mathbf{u},p)=\rho\Big(\frac{\partial\mathbf{u}}{\partial t}+(\mathbf{u}\cdot\nabla)\mathbf{u}\Big)+\nabla p-\nabla\cdot\big(2\mu\varepsilon(\mathbf{u})\big)-\mathbf{f}} ) \end{center} \end{document}

Residual-Based Variational Multiscale (RBVMS) Formulation for Incompressible Flow

Part II: Functional Scale Decomposition

The Variational Multiscale (VMS) method recognizes that the finite element space cannot represent every scale of the exact solution.

The exact velocity belongs to a Hilbert space:

[ \mathbf{u} \in V ]

and pressure:

[ p \in Q ]

The finite element approximation spaces are:

[ V_h \subset V ]

[ Q_h \subset Q ]

Because the finite element space is finite dimensional:

[ V_h \neq V ]

the exact solution contains unresolved components.

---

Orthogonal Decomposition

The VMS decomposition is:

[ V=\bar V \oplus V' ]

[ Q=\bar Q \oplus Q' ]

where:

• (\bar V): resolved finite element scales
• (V'): unresolved fine scales
• (\bar Q): resolved pressure scales
• (Q'): unresolved pressure scales

The fields are decomposed as:

[ \mathbf u=\bar{\mathbf u}+\mathbf u' ]

[ p=\bar p+p' ]

---

Cylinder Flow Governing Equations

The incompressible Navier-Stokes equations are:

[ \rho \left( \frac{\partial\mathbf u}{\partial t} + (\mathbf u\cdot\nabla)\mathbf u \right)

-\nabla p + \nabla\cdot (2\mu\epsilon(\mathbf u)) + \mathbf f ]

with continuity:

[ \nabla\cdot\mathbf u=0 ]

where

[ \epsilon(\mathbf u)

\frac12 (\nabla\mathbf u+\nabla\mathbf u^T) ]

is the strain-rate tensor.

---

Coarse and Fine Scale Problems

The exact problem can be written as two coupled variational problems.

Coarse-scale problem

Find:

[ (\bar{\mathbf u},\bar p) ]

such that:

[ \mathcal A ( \bar{\mathbf u}+\mathbf u', \bar p+p'; \bar{\mathbf w},\bar q )=0 ]

for all coarse-scale test functions.

---

Fine-scale problem

Find:

[ (\mathbf u',p') ]

such that:

[ \mathcal A ( \bar{\mathbf u}+\mathbf u', \bar p+p'; \mathbf w',q' )=0 ]

for all fine-scale test functions.

At this point no approximation has been introduced.

---

Part III: Fine-Scale Problem Derivation

Navier-Stokes Operator

Define:

[ \mathcal N(\mathbf u,p)

\rho \left( \frac{\partial\mathbf u}{\partial t} + (\mathbf u\cdot\nabla)\mathbf u \right) + \nabla p

\nabla\cdot(2\mu\epsilon(\mathbf u))

\mathbf f ]

The exact solution satisfies:

[ \mathcal N(\mathbf u,p)=0 ]

---

Fréchet Linearization

Using:

[ \mathbf u=\bar{\mathbf u}+\mathbf u' ]

and

[ p=\bar p+p' ]

we obtain:

[ \mathcal N ( \bar{\mathbf u}+\mathbf u', \bar p+p' )

\mathcal N(\bar{\mathbf u},\bar p) + D\mathcal N [ \mathbf u',p' ] + O(||\mathbf u'||^2) ]

---

Coarse-Scale Residual

The momentum residual is:

[ \boxed{ \mathbf R_M

\rho \left( \frac{\partial\bar{\mathbf u}}{\partial t} + (\bar{\mathbf u}\cdot\nabla)\bar{\mathbf u} \right) + \nabla\bar p

\nabla\cdot (2\mu\epsilon(\bar{\mathbf u}))

\mathbf f } ]

The continuity residual is:

[ \boxed{ R_C= \nabla\cdot\bar{\mathbf u} } ]

---

Fine-Scale Differential Equation

The fine scales satisfy:

[ D\mathcal N [ \mathbf u',p' ]

-\mathbf R_M ]

The linearized operator is:

[ D\mathcal N[\mathbf u',p']

\rho \frac{\partial\mathbf u'}{\partial t} + \rho (\bar{\mathbf u}\cdot\nabla)\mathbf u' + \rho (\mathbf u'\cdot\nabla)\bar{\mathbf u} + \nabla p'

\nabla\cdot (2\mu\epsilon(\mathbf u')) ]

---

Operator Representation

Define:

[ \mathcal L=D\mathcal N ]

Then:

[ \boxed{ \mathcal L(\mathbf u',p')

-(\mathbf R_M,R_C) } ]

---

Green's Function Representation

The exact fine scales are:

[ \boxed{ (\mathbf u',p')

-\mathcal L^{-1} (\mathbf R_M,R_C) } ]

The inverse operator is replaced by a local approximation:

[ \mathcal L^{-1} \approx \boldsymbol{\tau} ]

giving:

[ \boxed{ (\mathbf u',p')

-\boldsymbol{\tau} (\mathbf R_M,R_C) } ]

or:

[ \mathbf u'

-\tau_M\mathbf R_M ]

and

[ p'

-\tau_C R_C ]

---

Physical Interpretation

The unresolved scales are not empirical turbulence fluctuations.

They are the response of the unresolved solution space to the residual of the resolved finite element solution.

Large residual:

[ \Rightarrow \text{large unresolved scales} ]

Small residual:

[ \Rightarrow \text{small unresolved scales} ]

This is the fundamental difference between RBVMS and traditional turbulence models based on prescribed eddy viscosity.

---

Next Development Stage

The next step is the derivation of the stabilization tensor:

[ \boldsymbol{\tau} ]

and specifically:

[ \tau_M ]

and

[ \tau_C ]

from the characteristic time scales of:

1. Temporal evolution

[ \frac{1}{\Delta t} ]

2. Convection

[ \frac{|\mathbf u|}{h} ]

3. Diffusion

4. $$ \mathcal{N}(\mathbf{u},p) = \rho \left( \frac{\partial\mathbf{u}}{\partial t}

(\mathbf{u}\cdot\nabla)\mathbf{u} \right) + \nabla p

\nabla\cdot (2\mu\epsilon(\mathbf{u}))

\mathbf f $$

[ \frac{\nu}{h^2} ]

These parameters will lead directly to the final RBVMS weak formulation and the C++ finite element implementation. Some text describing the equation.

$$ \mathcal{N}(\mathbf{u},p)

\rho \left( \frac{\partial\mathbf{u}}{\partial t} + (\mathbf{u}\cdot\nabla)\mathbf{u} \right) + \nabla p

\nabla\cdot (2\mu\epsilon(\mathbf{u}))

\mathbf{f} $$

More text after the equation.