Probabilistic Solution of the Navier-Stokes Equations on a 2D Torus
April 23, 2014
1
NS Equations
The Navier-Stokes (NS) PDEs model the time evolution of P components of the velocity, u : D →
RP , of an incompressible fluid on a spatial domain X . The NS boundary value problem on the
spatio-temporal domain D = X × T is given by:

(
)
∂


∂t u − θ ∆u + u · ∇ u


 ∇·u
∫ (j)

u dx



 u
= f − ∇p,
(x, t) ∈ D,
= 0,
(x, t) ∈ D,
= 0,
(x, t) ∈ D, j = 1, 2,
= uB ,
(x, t) ∈ X × {0},
(1)
where
(
∆u =
∂2
∂x(1)
2 u1
+
∂2
∂x(2)
2 u1 ,
∂2
∂x(1)
2 u2
+
∂2
∂x(2)
)⊤
2 u2
=
2
∑
m=1
(
2
∑
∂2
m′ =1
∂x(m′ )
)
2
em u
and
(
)
( 2
(
)⊤ ∑
2
∑
∂
∂
∂
∂
∂
(m′ )
u · ∇ ũ = u1 (1) ũ1 + u2 (2) ũ1 , u1 (1) ũ2 + u2 (2) ũ2
u
em ũ(m)
=
(m′ )
∂x
∂x
∂x
∂x
∂x
′
m=1
)
m =1
This model is parameterized by the viscosity of the fluid, θ > 0; the pressure function p : D → R;
and the external time-homogeneous forcing function f : X → R. We consider the NS equations over
a 2-dimensional torus shaped domain, X = [0, 2π) × [0, 2π), expressed in spherical coordinates. We
further assume periodic boundary conditions (second and third lines of equation 1), and viscosity
θ = 1 ×10−3 in the turbulent regime. For simplicity, we consider the unforced NS equations (f = 0).
Often, the quantity of interest is the vorticity, or local spinning motion of the incompressible
fluid, which we define as,
ϖ(x, t) = −∇ × u(x, t),
where clockwise rotation corresponds to positive vorticity. Vorticity can also help to better visualize
the solution of the NS system by summarizing the two components of velocity by a one-dimensional
1
function.
2
Spectral projection for NSE
The phase space of the solution to (1) at a given time point is the Sobolev space,
{
H := v ∈ L2 (X ) :
∂
v ∈ L2 (X ), ∇ · v = 0,
∂x(i)
∫
v (j) dx = 0
}
and let H be its closure. This is a subspace of L2 , endowed with the L2 inner product. Also define
(
)2
P : L2 (X ) → H. We want to project (1) to the subspace H. Now we can rewrite (1) as
du − θ Au + B (u, u) = −P + f, u(0) = v
0
∂t
(2)
where P is the solution to the Poisson equation,
(
)
∇· u·∇ u=−
2
∑
∂
∂x(m)
m=1
(
2
∑
u
(m′ )
m′ =1
∂
∂x(m′ )
)
u(m) = −
2
∑
m=1
2
∑
∂2
∂x(m)
∂2
(m)
m=1 ∂x
2p
2p
Next, we replace u(x, t) in (2) by
u(x, t) = ûk (t) ϕk (x),
were ûk (t) = u(x, t) ϕk (x),
k = Z2 \{0},
and hereafter omit spatial and temporal dependence in the Fourier modes and coefficients from the
notation. We then project (2) on the Fourier bases to obtain,
(j)
û˙ ℓ = −θ
2 (
2
)
∑
∑
(m) 2 (j)
(m) (m) (j)
(j)
wℓ
ûℓ − i
ûk wℓ−k ûℓ−k + fˆℓ + ∑
2
m=1
1
∑
(m) 2
m,m′
m=1 wℓ
m=1
(m′ )
(m)
ûℓ−k wℓ
(m′ ) (m)
ûk
wk
(3)
with initial condition
(j)
(j)
ûℓ (0) = ⟨v, ϕℓ ⟩ = v̂ℓ .
2.1
Derivation of expression (3)
The Fourier bases,
(j)
(1)
(2)
ϕk := exp(iwk x(1) ) exp(iwk x(2) ) = exp(iwk · x),
2
with wavenumbers,
2
∑
−1
wk = (2π)
em k,
k ∈ Z2 \{0},
m=1
form an orthonormal basis for H, where ej is the unit vector in the jth direction We will use the
partial derivatives of these Fourier modes,
∂
(j)
(m) (j)
(m)
ϕ = iwk exp(iwk · x) = iwk ϕk
∂x(m) k
(
)
∂ 2 (j)
(m) 2
(m) (j)
ϕ
=
−
w
exp(iwk · x) = −wk ϕk .
k
∂x(m) k
and the orthonormality property,
{
(j)
(j)
⟨ϕk , ϕℓ ⟩ =
0 if k ̸= ℓ;
1 if k = ℓ.
where ⟨·, ·⟩ represents the L2 norm. We also use the fact that
{
(j) (j)
⟨ϕk ϕk′ , ϕℓ ⟩
=
0 if k + k′ ̸= ℓ;
1 if k + k′ = ℓ.
We need to project (2) to the ℓth Fourier mode. Let us do this for each term of the equation
separately. Denote by the subscript j the jth component of each vector. Therefore,
)
(
d (j)
∂ (j) (j)
(j) (j)
ϕk , ϕ ℓ ⟩
û
⟨ u , ϕℓ ⟩ = ⟨
∂t
dt k
d (j) (j) (j)
= ûk ⟨ϕk , ϕℓ ⟩
dt
d (j)
(j)
= ûℓ := û˙ ℓ
dt
The projection of the forcing function is just its Fourier transform,
(j)
(j)
⟨f (j) , ϕℓ ⟩ = fˆℓ
The projection of the jth component of the Laplacian A on the ℓth Fourier mode is,
⟨Au
(j)
(j)
, ϕℓ ⟩
=⟨
2
∑
∂2
m=1
2
∂x(m)
3
(j)
u(j) , ϕℓ ⟩
= ⟨−
2 (
)
∑
(m) 2 (j) (j) (j)
wk
ûk ϕk , ϕℓ ⟩
m=1
2 (
)
∑
(j) (j)
(m) 2 (j)
=−
wk
ûk ⟨ϕk , ϕℓ ⟩
=−
m=1
2 (
∑
)
(m) 2
wℓ
(j)
ûℓ
m=1
The projection of the jth component of B on the ℓth Fourier mode is,
(j)
⟨B(u, u(j) ), ϕℓ ⟩ = ⟨
2
∑
m=1
2
∑
=⟨
=i
=i
m=1
2
∑
m=1
2
∑
u(m)
∂
(j)
u(j) , ϕℓ ⟩
∂x(m)
(m) (m)
(m) (j) (j)
(j)
ûk ϕk
iwk′ ûk′ ϕk′ , ϕℓ ⟩
(m)
(m) (j)
(m) (j)
(m)
(m) (j)
(j)
ûk wk′ ûk′ ⟨ϕk ϕk′ , ϕℓ ⟩
ûk wℓ−k ûℓ−k
m=1
where the second to last line uses the fact that k′ = ℓ − k. In order to compute the projection of
the jth component of pressure P , we first note that the assumption of a constant body force and
divergence free condition is satisfied by letting
2
∑
∂
∂x(m)
m=1
(
2
∑
u
(m′ )
m′ =1
∂
∂x(m′ )
)
u(m) = −
2
∑
∂2
m=1
∂x(m)
2p
we then project both sides to the ℓth Fourier mode as follows:
2
∑
∂
⟨
∂x(m)
m=1
(
2
∑
m′ =1
u
(m′ )
∂
∂x(m′ )
)
(j)
u(m) , ϕℓ ⟩
( 2
)
∑ (m′ ) (m′ ) (m′ ) (m) (m)
∂
(j)
= ⟨i
ûk′ ϕk′ wk ûk ϕk
, ϕℓ ⟩
(m)
∂x
m=1
m′ =1
∑ (m′ ) (m′ ) (m) ∂
(m′ ) (m) (j)
ϕk ′ ϕk , ϕ ℓ ⟩
= ⟨i
ûk′ wk ûk
(m)
∂x
m,m′
)
∑ (m′ ) (m′ ) (m) ( (m′ ) (m′ ) (m)
(m) (m′ ) (m)
(j)
= ⟨−
ûk′ wk ûk
wk′ ϕk′ ϕk + wk ϕk′ ϕk
, ϕℓ ⟩
2
∑
m,m′
4
=−
∑
(m′ )
(m′ ) (m)
ûk
(
)
(m′ ) (m′ ) (m) (j)
(m) (m′ ) (m) (j)
wk′ ⟨ϕk′ ϕk , ϕℓ ⟩ + wk ⟨ϕk′ ϕk , ϕℓ ⟩
(m′ )
(m′ ) (m)
ûk
(
)
(m′ )
(m)
wℓ−k + wk
(m′ )
(m)
ûk′ wk
m,m′
=−
∑
ûℓ−k wk
m,m′
=−
∑
ûℓ−k wℓ
(m′ ) (m)
ûk
wk
m,m′
where the second to last line uses the fact that k′ = ℓ − k. And,
⟨−
2
∑
m=1
∂2
∂x
(j)
p, ϕℓ ⟩
(m) 2
=⟨
2
∑
(m) 2
wk
m=1
2
∑
= p̂ℓ
(j)
(j)
p̂k ϕk , ϕℓ ⟩
(
)
(m) 2
wℓ
m=1
We can then equate these expressions and solve for p̂ℓ :
−
∑
(m′ ) (m) (m′ ) (m)
ûℓ−k wℓ wk ûk
m,m′
= p̂ℓ
2
∑
(m) 2
wℓ
m=1
∑ (m′ ) (m) (m′ ) (m)
1
p̂ℓ = − ∑
ûℓ−k wℓ wk ûk
(
)2
(m)
2
′
m,m
m=1 wℓ
3
Discretization in space
The Navier-Stokes model (1) was reduced to a set of 128 × 128 stiff coupled ODEs with associated
constraints through a pseudo-spectral projection in Fourier space, described above. The spatial
discretization grid was equally spaced. The initial velocity field is generated from a bivariate
normal distribution at each of the mesh points.
4
Exponential time differencing
Since each of the 128 × 128 ODEs have both linear and nonlinear terms, the system is stiff and
the numerical solutionwill require prohibitively small time-steps. To overcome this problem, we
use exponential time-differencing to solve the linear part of each ODE exactly, leaving only the
nonlinear part. Note that the projected equation (3) can be written as follows:
5
(
(j)
û˙ ℓ
−θ
=
2 (
∑
)
(m) 2 (j)
wℓ
ûℓ
)
+
(j)
fˆℓ
m=1


+ −i
2
∑
m=1

(m)
(m)
∑ (m′ ) (m) (m′ ) (m) 
1
ûℓ−k wℓ wk ûk 
(
)2
(m)
′
w
m,m
m=1
ℓ
(j)
ûk wℓ−k ûℓ−k + ∑
2
(j)
:= Lûℓ + N (û(ℓ−k) , ûk ; t),
Therefore we can apply the method of exponential time-differencing by using the following transformation at each step:
w = exp{−Lt} u
which gives transforms the system to the following completely nonlinear ODE,
d
w = exp{−Lt} N (w; t)
dt
6