Spectral representation of an incompressible flow

[Wuberdall]

Hi PH.

Let $u_i(\mathbf{x},t)$ be the velocity field in a periodic box of linear size $2\pi$. The spectral representation of $u_i(\mathbf{x},t)$ is then
$$u_i(\mathbf{x},t) = \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$$ where ι denotes the usual imaginary unit.

A flow is incompressible iff $u_i(\mathbf{x})$ is a solenodial verctor field, that is
$$\nabla_iu_i(\mathbf{x},t) = 0$$
Combining the above, I get
$$\begin{aligned}0 &= \nabla_iu_i(\mathbf{x},t) \\
&= \sum_{\mathbf{k}\in\mathbb{Z}^3}\hat{u}_i(\mathbf{k},t)\nabla_ie^{\iota k_jx_j} \\
&= \sum_{\mathbf{k}\in\mathbb{Z}^3}\iota k_j\delta_{ij}\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} \\
&= \iota\sum_{\mathbf{k}\in\mathbb{Z}^3}k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}\end{aligned}$$

If $u_i(\mathbf{x},t)$ is to represent a physical flow velocity field, then $u_i(\mathbf{x},t)$ must be real function. That is $\mathbf{u}(\mathbf{x}):[0,2\pi]^3\times\mathbb{R}\rightarrow\mathbb{R}^3$ and thus $\hat{u}_i(\mathbf{k}) = \hat{u}^\ast_i(-\mathbf{k})$. Therefore
$$\begin{aligned} \iota\sum_{\mathbf{k}\in\mathbb{Z}^3}k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} &= -\iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\hat{u}_i(-\mathbf{k},t)e^{-\iota k_jx_j} + 0 + \iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} \\
&= \iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\big[\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} - \hat{u}_i(-\mathbf{k},t)e^{-\iota k_jx_j}\big] \\
&= \iota\sum_{\mathbf{k}\in\mathbb{N}^3}k_j\big[\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j} - \hat{u}^\ast_i(\mathbf{k},t)\big(e^{\iota k_jx_j}\big)^\ast\big] \\
&= 2\iota\sum_{\mathbf{k}\in\mathbb{N}^3}\Im\big[k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}\big] \end{aligned}$$

From the harmonic analysis I would expect the incompressibility condition to be $$k_i\hat{u}_i(\mathbf{k},t) = 0$$
Clearly, if this is true the incompressibility condition would be true. But I can't see how (unless I assume a sort of detailed balance in the above sum) the incompressibility condition imply the orthogonal relation $k_i\hat{u}_i(\mathbf{k},t) = 0$.

Any help is appreciated :-)

 

[AndyW]

Hello,

Thank you for sharing your thoughts and equations on the spectral representation of velocity fields in a periodic box. Your approach seems to be correct, and I agree that the incompressibility condition should imply the relation $k_i\hat{u}_i(\mathbf{k},t) = 0$. Here are a few points that might help clarify the connection between the two:

1. The incompressibility condition implies that the velocity field is divergence-free, meaning that the net flow into any small region is zero. This can be seen from the continuity equation, which states that the rate of change of density at any point is equal to the divergence of the velocity field at that point. If the velocity field is divergence-free, then the rate of change of density is zero and the density remains constant.

2. In the spectral representation, the term $k_j\hat{u}_i(\mathbf{k},t)e^{\iota k_jx_j}$ can be interpreted as a wave with wavenumber $k_j$ in the $j^{th}$ direction. This means that the flow at each point in space can be decomposed into a sum of waves with different wavenumbers and amplitudes. If the flow is incompressible, then the net flow into any small region must be zero, which means that the waves with positive wavenumbers must cancel out the waves with negative wavenumbers. This is only possible if the amplitudes of these waves satisfy the relation $k_i\hat{u}_i(\mathbf{k},t) = 0$.

3. The incompressibility condition is a necessary condition for a physical flow, but it is not sufficient. In order for a flow to be physically meaningful, it must also satisfy the Navier-Stokes equations, which describe the conservation of momentum and the effects of viscosity. These equations can be solved for the velocity field in terms of the pressure field, and it can be shown that the incompressibility condition is automatically satisfied if the pressure field is a function of only the position and not time. This is known as the incompressible Navier-Stokes equations.

I hope this helps clarify the connection between the incompressibility condition and the relation $k_i\hat{u}_i(\mathbf{k},t) = 0$. If you have any further questions or thoughts, please feel free to share them. Best