Solution to the Periodic Navier-Stokes Equation

August 13, 2016

Immortal Matrix Exponential Solution

Filed under: Mathematics — David Purvance @ 2:11 pm

The three-dimensional periodic Navier-Stokes equation for incompressible flows is posed as a nonlinear matrix differential equation. A general solution is developed in the form of a power series in time having vector coefficients that are a function of wavenumber alone. This general solution is argued, but not proven, to exist only for finite time intervals.  The matrix exponential solution is then introduced. It requires commutative coefficient matrices, constructed in wavenumber space by assuming self-similar vector coefficients that remain parallel to the initial flow for all time orders. This self similarity study is motivated by the discovery thru numerical simulations that vector coefficients for successive time orders become highly correlated and therefore exhibit a self-similar behavior.

1.0 The Periodic Navier-Stokes Matrix Differential Equation

Begin with the {\mathbb{C}^{3{\text{x}}1}} Navier-Stokes equation [1] describing the time evolution of an incompressible flow’s M Fourier modes {\mathbf{u}}\left( {{{\boldsymbol{\kappa }}_j},t} \right)

\left(d/dt+\nu \kappa _{j}^{2} \right) \mathbf{u}\left( {{\boldsymbol{\kappa }}_{j}},t \right) =-P\left({{\boldsymbol{\kappa }}_{j}}\right)\sum\limits_{k=1}^{M}{J\left( \mathbf{u},{{\boldsymbol{\kappa }}_{j}},{{{\boldsymbol{{\kappa }'}}}_{k}},t \right)\,\mathbf{u}\left( {{{\boldsymbol{{\kappa }'}}}_{k}},t \right)} (1)

where \boldsymbol{\kappa}_j and \boldsymbol{\kappa}'_k are M={{\left( L+1 \right)}^{3}} discrete wavenumbers \left( {{l}_{1}},{{l}_{2}},{{l}_{3}} \right)d\kappa with {{l}_{1}},{{l}_{2}},{{l}_{3}}=\left( -L/2,...,0,..,L/2 \right) for some even integer L and where d\kappa ={2\pi }/{L}. Included in (1) are \kappa _{j}^{2}={{\left| {{\boldsymbol{\kappa }}_{j}} \right|}^{2}}, a known viscosity \nu \ge 0, the incompressibility condition {{\boldsymbol{\kappa }}_{j}}\cdot \mathbf{u}\left( {{\boldsymbol{\kappa }}_{j}},t \right)=0, the projection tensor

P\left( {{{\boldsymbol{\kappa }}_j}} \right) = \left[ {\begin{array}{*{20}{c}}{1 - \frac{{\kappa _{j,1}^2}}{{\kappa _j^2}}}&{ - \frac{{{\kappa _{j,1}}{\kappa _{j,2}}}}{{\kappa _j^2}}}&{ - \frac{{{\kappa _{j,1}}{\kappa _{j,3}}}}{{\kappa _j^2}}}\\{ - \frac{{{\kappa _{j,2}}{\kappa _{j,1}}}}{{\kappa _j^2}}}&{1 - \frac{{\kappa _{j,2}^2}}{{\kappa _j^2}}}&{ - \frac{{{\kappa _{j,2}}{\kappa _{j,3}}}}{{\kappa _j^2}}}\\{ - \frac{{{\kappa _{j,3}}{\kappa _{j,1}}}}{{\kappa _j^2}}}&{ - \frac{{{\kappa _{j,3}}{\kappa _{j,2}}}}{{\kappa _j^2}}}&{1 - \frac{{\kappa _{j,3}^2}}{{\kappa _j^2}}}\end{array}} \right], (2)

and

J\,\left( {{\bf{u}},{{\boldsymbol{\kappa }}_j},{{{\boldsymbol{\kappa '}}}_k},t} \right) = \left[ {\begin{array}{*{20}{c}}{i{\kappa _{j,1}}{u_1}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}&{i{\kappa _{j,2}}{u_1}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}&{i{\kappa _{j,3}}{u_1}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}\\{i{\kappa _{j,1}}{u_2}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}&{i{\kappa _{j,2}}{u_2}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}&{i{\kappa _{j,3}}{u_2}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}\\{i{\kappa _{j,1}}{u_3}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}&{i{\kappa _{j,2}}{u_3}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}&{i{\kappa _{j,3}}{u_3}\left( {{{\boldsymbol{\kappa }}_j} - {{{\boldsymbol{\kappa '}}}_k},t} \right)}\end{array}} \right]. (3)

Simultaneously consider all wavenumbers {{\boldsymbol{\kappa }}_j} by writing (1) in consecutive trios of rows for j = 1,...,M. This forms the periodic Navier-Stokes matrix differential equation

d{\bf{u}}/dt = U\left( {\bf{u}} \right){\bf{u}} (4)

where

{\bf{u}} = \left[ {\begin{array}{*{20}{c}}{{\bf{u}}\left( {{{\boldsymbol{\kappa }}_1},t} \right)}\\\vdots\\{{\bf{u}}\left({{{\boldsymbol{\kappa }}_M},t} \right)}\end{array}}\right] = \left[ {\begin{array}{*{20}{c}}{{\bf{u}}\left( {{{{\boldsymbol{\kappa '}}}_1},t} \right)}\\\vdots\\{{\bf{u}}\left( {{{{\boldsymbol{\kappa '}}}_M},t} \right)}\end{array}} \right] (5)

and

U\left( {\bf{u}} \right) = D + PJ\left( {\bf{u}} \right), (6)

with

D = \left[ {\begin{array}{*{20}{c}}{D\left( {\kappa _1^2} \right)}&\cdots &0\\ \vdots & \ddots & \vdots\\ 0& \cdots &{D\left( {\kappa _M^2} \right)}\end{array}}\right], (7)

D\left( {\kappa _j^2} \right) = \left[ {\begin{array}{*{20}{c}}{ - \nu \kappa _j^2}&0&0\\ 0&{ - \nu \kappa _j^2}&0\\ 0&0&{ - \nu \kappa _j^2}\end{array}} \right], (8)

P = \left[ {\begin{array}{*{20}{c}}{- P\left( {{{\boldsymbol{\kappa }}_1}} \right)}& \cdots &0\\ \vdots & \ddots & \vdots\\ 0& \cdots &{ - P\left( {{{\boldsymbol{\kappa }}_M}}\right)}\end{array}}\right] (9)

and

J\left( {\mathbf{u}} \right) = \left[ {\begin{array}{*{20}{c}}{J\left( {{\mathbf{u}},{{\boldsymbol{\kappa }}_1},{{{\boldsymbol{\kappa '}}}_1},t} \right)}& \cdots &{J\left( {{\mathbf{u}},{{\boldsymbol{\kappa }}_1},{{{\boldsymbol{\kappa '}}}_M},t}\right)}\\ \vdots & \ddots & \vdots\\ {J\left( {{\mathbf{u}},{{\boldsymbol{\kappa }}_M},{{{\boldsymbol{\kappa '}}}_1},t}\right)}& \cdots &{J\left( {{\mathbf{u}},{{\boldsymbol{\kappa }}_M},{{{\boldsymbol{\kappa '}}}_M},t} \right)}\end{array}}\right]. (10)

2.0 Power Series Solution

Expand \mathbf{u}\left( {{\boldsymbol{\kappa }}_{j}},t \right) as a time series

\mathbf{u}\left( {{\boldsymbol{\kappa }}_{j}},t \right)=\sum\limits_{n=0}^{\infty }{{{\mathbf{u}}_{n}}\left( {{\boldsymbol{\kappa }}_{j}} \right)}\ {{t}^{n}} (11)

with unknown coefficients {{\mathbf{u}}_{n}}\left( {{\boldsymbol{\kappa }}_{j}} \right) and more simply {{\mathbf{u}}_{n}}. Substituting (11) into (1) and performing the time differentiation gives

\sum\limits_{n=0}^{\infty }{n{{\mathbf{u}}_{n}}{{t}^{n-1}}}=\left( \sum\limits_{p=0}^{\infty }{{{U}_{p}}{{t}^{p}}} \right)\sum\limits_{q=0}^{\infty }{{{\mathbf{u}}_{q}}{{t}^{q}}}=\sum\limits_{p=0}^{\infty }{\sum\limits_{q=0}^{\infty }{{{U}_{p}}{{\mathbf{u}}_{q}}{{t}^{p+q}}}} (12)

with

{{U}_{p}}={{\delta }_{0.p}}D+P{{J}_{p}}. (13)

Matching coefficients in (12) when {{t}^{n-1}}={{t}^{p+q}} gives a recursion formula for the unknown flow coefficients

{{\mathbf{u}}_{n}}=\frac{1}{n}\sum\limits_{p=0}^{n-1}{{{U}_{p}}}{{\mathbf{u}}_{n-1-p}}. (14)

It begins with a known initial flow \mathbf{u}\left( {{\mathbf{\kappa }}_{j}},t=0 \right)={{\mathbf{u}}_{0}} with an assumed magnitude of unity. As M\to \infty equations (11) and (14) describe the continuous time series solution to the periodic Navier-Stokes equation.

3.0 The Recursion Formula

Reducing the dimensionality of the quantities in (14) to a scalar, beginning with x_0 and ignoring the wavenumber factors quickly reveals that this recursion formula creates coefficients geometric in nature {{x}_{0}},{{x}_{0}}\cdot {{x}_{0}},{{x}_{0}}\cdot {{x}_{0}}\cdot {{x}_{0}},...,x_{0}^{n}, and therefore time series coefficients admitting only a finite time interval of convergence. Extensive numerical simulations evaluating \left| \sum\limits_{n=0}^{N}{{{\mathbf{u}}_{n}}{{t}^{n}}} \right| for increasing values of N, using (14) and mostly white-noise \mathbf{w} water flows {{\mathbf{u}}_{0}}=P\mathbf{w} with a paltry L=8 and without the no-slip boundaries, corroborate this universally-predicted, power-law blow-up behavior. This is far from a proof though.

4.0 The Immortal Matrix Exponential Solution

There is a way out of this apparent hopelessness.  It involves restricting wavenumber portions of the solution to self-similar vector coefficients over all time orders and parallel to the initial flow. Numerical simulations show that solution coefficient vectors for successive time orders are highly correlated.

As a means of its mathematical introduction, notice that back substituting in (14) for {{\mathbf{u}}_{n}} and keeping just the first term involving {{U}_{0}}{{\mathbf{u}}_{n-1}} etc., yields

{{\mathbf{u}}_{n}}=\frac{1}{n}{{U}_{0}}{{\mathbf{u}}_{n-1}}=\frac{1}{n}\frac{1}{n-1}U_{0}^{2}{{\mathbf{u}}_{n-2}}=...=\frac{1}{n!}U_{0}^{n}{{\mathbf{u}}_{0}}, (15)

which when the time variable is added, represents the series expansion of \exp ({{U}_{0}}t){{\mathbf{u}}_{0}}.  This is stable for non-positive real eigenvalues which is certainly the case for viscous {D} and turns out to be the case for all convective P{{J}_{n}}. The convective eigenvalues are purely imaginary (or zero) and come in conjugate pairs. Its eigenvectors form a unitary basis. Neither are proven here.

The viscous part of (15) is well understood, but the convective part is not, and, along with (4) suggests the universally convergent matrix exponential solution [2]

\mathbf{u}=\exp \left( \sum\limits_{n=0}^{\infty }{\frac{1}{n+1}{{U}_{n}}{{t}^{n+1}}} \right){{\mathbf{u}}_{0}}=\prod\limits_{n=0}^{\infty }{\exp }\left( \frac{1}{n+1}{{U}_{n}}{{t}^{n+1}} \right){{\mathbf{u}}_{0}}. (16)

It arises when {{U}_{n}} commute. If

{{U}_{n}}={{V}_{n}}{{\Lambda }_{n}}V_{n}^{-1} (17)

is the eigendecomposition of {{U}_{n}}, commutativity requires that eigenvectors {{V}_{n}} are time-order independent

{{V}_{n}}={{V}_{0}}. (18)

In view of (3) and (10) one way of achieving (18) is to require self-similar {\mathbf{u}_{n}}

{{\mathbf{u}}_{n}}={{u}_{n}}{{\mathbf{u}}_{0}}, (19)

for scalar {{u}_{n}}.

Constraint (19) requires that the projection matrix P project flow vectors parallel to initial flow rather than just onto the plane perpendicular to wavenumber. This can be achieved by

{{\mathbf{u}}_{n}}=\mathbf{u}_{0}^{T}\left( \frac{1}{n}\sum\limits_{p=0}^{n-1}{\left( {{\delta }_{0,p}}D+{{J}_{p}} \right){{\mathbf{u}}_{n-1-p}}} \right){{\mathbf{u}}_{0}} (20)

rather than (14). Most importantly, (20) maintains incompressibility because the initial flow by assumption is incompressible.

Remarks

The above sections describe a purely mathematical invention and construction of a regular solution to the periodic Navier-Stokes equation (1), and, as such, should stand on its own merit. Is it universal? No, because it assumes self-similar wavenumber solution coefficient vectors that are parallel to the initial flow. It simply explains the required assumption for the matrix exponential solution. It is motivated by the discovery through numerical simulations that solution coefficient vectors are highly correlated.

References

[1] Pope SB 2003 Turbulent Flows eq 6.146 Cambridge University Press NY NY [2] arxiv.org/abs/math/0610086

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