Solution to the Periodic Navier-Stokes Equation

August 13, 2016

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 and matrix 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 directionally stationary flow coefficients that remain parallel to the initial flow. Directional stationarity is argued by noting that a flow’s pressure gradient at each wavenumber is directionally stationary. While directional stationarity in wavenumber space does force directional stationarity in physical space OVER TIME, this solution at this point remains nothing more than a curiosity.

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 attention to directionally stationary flows, i.e., flows that in wavenumber space are constant in direction but varying in magnitude over time. It keys off the observation that a flow’s pressure gradient is always parallel to wavenumber (see [1] for example) and therefore is directionally stationary. This restriction, however, does not force directional stationarity in physical space.

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}}. These are directionally stationary flows, flows that can vary wildly in magnitude and direction from wavenumber to wavenumber but are constant in direction at each wavenumber. Directional stationarity in wavenumber does NOT imply directional stationarity in physical space.

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. In wavenumber space it requires that flows remain directionally stationary as does pressure gradient. While this does not force directional stationarity in physical space, and, therefore permits flows that can vary in magnitude and direction, it remains just a curiosity.

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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