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1. What the student is asking (plain‑language restatement)

The student has already measured the 3‑D energy spectrum of a homogeneous, isotropic turbulent field by

  1. taking a 3‑D Fourier transform of the velocity data (periodic in x, y, z),
  2. binning the Fourier coefficients by the magnitude of the wavenumber k, and
  3. obtaining the familiar (E(k)\sim k^{-5/3}) inertial‑range law.

Now the student wants to do the same for a turbulent boundary layer where the flow is anisotropic and non‑periodic in the wall‑normal direction (let us call it (y)).

The questions are:

  1. How can we compute an energy spectrum for a fully three‑dimensional turbulent boundary layer?
  2. If a 3‑D spectrum is impossible, how do we define “scales” (or a spectrum) in the wall‑normal direction?

In other words: What mathematical tools replace the simple 3‑D FFT when one direction is bounded, and how do we interpret the resulting spectra?

Below is a complete, step‑by‑step guide that answers both points.

2. Step‑by‑step solution

2.1 Review: Energy spectrum in a homogeneous, isotropic flow

For a velocity field (\mathbf{u}(\mathbf{x})) defined in a periodic box of size (L_x\times L_y\times L_z),

  1. Compute the discrete Fourier transform (DFT)

    [ \hat{\mathbf{u}}(\mathbf{k}) = \frac{1}{N_x N_y N_z} \sum_{n_x=0}^{N_x-1}!!\sum_{n_y=0}^{N_y-1}!!\sum_{n_z=0}^{N_z-1} \mathbf{u}(x_{n_x},y_{n_y},z_{n_z})\, e^{-i\mathbf{k}\cdot\mathbf{x}}, ]

    where (\mathbf{k}=(k_x,k_y,k_z)=\bigl(2\pi m_x/L_x,2\pi m_y/L_y,2\pi m_z/L_z\bigr)).

  2. The spectral energy density (per unit wavenumber) is

    [ E(\mathbf{k}) = \frac12\bigl|\hat{\mathbf{u}}(\mathbf{k})\bigr|^{2}. ]

  3. Because the flow is isotropic, one can collapse the 3‑D data into a 1‑D spectrum by binning all modes that have the same wavenumber magnitude

    [ k = |\mathbf{k}| = \sqrt{k_x^{2}+k_y^{2}+k_z^{2}} . ]

    The binned value

    [ E(k) = \sum_{k\le|\mathbf{k’}|<k+\Delta k} E(\mathbf{k’}). ]

  4. In the inertial range, (E(k) \propto \varepsilon^{2/3}k^{-5/3}) (Kolmogorov).

All of the above relies on periodicity (or at least statistical homogeneity) in the three directions. In a turbulent boundary layer, only the streamwise ((x)) and spanwise ((z)) directions are (approximately) homogeneous; the wall‑normal direction ((y)) is bounded by the wall and by the free stream, so we cannot use a plain FFT there.

2.2 What can we still do? – Use the two homogeneous directions

Because the flow is statistically homogeneous in (x) and (z), we can still Fourier‑transform in those two directions exactly as before:

[ \hat{\mathbf{u}}(k_x, y, k_z) = \frac{1}{N_x N_z} \sum_{n_x=0}^{N_x-1}!!\sum_{n_z=0}^{N_z-1} \mathbf{u}(x_{n_x},y, z_{n_z})\, e^{-i(k_x x_{n_x}+k_z z_{n_z})}. ]

The result is a 2‑D spectral field that still depends on the wall‑normal coordinate (y). For each pair ((k_x,k_z)) we have a complex vector that varies with (y).

2.2.1 2‑D (streamwise–spanwise) energy spectrum at a given wall‑normal location

Define the local spectral density

[ E(k_x,k_z; y) = \frac12 \bigl| \hat{\mathbf{u}}(k_x, y, k_z) \bigr|^{2}. ]

If you want a one‑dimensional spectrum at a fixed wall‑normal plane (e.g. at (y=y_0)), you can collapse the 2‑D data in the usual way:

[ E(k; y_0)=\sum_{\sqrt{k_x^{2}+k_z^{2}} \in [k,\,k+\Delta k]} E(k_x,k_z; y_0), ]

which yields a planar energy spectrum that is still expected to follow (k^{-5/3}) over the inertial range (provided (y_0) is not too close to the wall).

2.3 Extending to a 3‑D spectrum: treat the wall‑normal direction with a non‑periodic basis

If we truly want a 3‑D spectral representation, we must replace the Fourier series in (y) by a set of orthogonal functions that respect the boundary conditions:

Boundary condition Suitable basis Remarks
No‑slip wall at (y=0) and free‑stream (or symmetry) at (y=h) Chebyshev polynomials (type‑I or type‑II) or sine/cosine series with appropriate parity Common in DNS of channel / boundary‑layer flow
Wall‑bounded, non‑periodic, but still homogeneous in the statistical sense Discrete Cosine Transform (DCT) (even extension) or Discrete Sine Transform (DST) (odd extension) Simpler to implement than Chebyshev, retains orthogonality on a uniform grid
Complex geometry (curved wall) Proper Orthogonal Decomposition (POD) or Dynamic Mode Decomposition (DMD) Gives a data‑driven modal basis; not a wavenumber spectrum per se, but a scale‑decomposition

Below we outline the Chebyshev‑spectral approach because it is the most widely used in wall‑bounded DNS.

2.3.1 Chebyshev expansion in (y)

  1. Map the physical wall‑normal interval ([0,\,\delta]) (or ([0,\,h])) to the Chebyshev domain ([-1,\,1]) via

    [ \eta = 2\frac{y}{\delta} - 1 . ]

  2. Expand the wall‑normal dependence of each Fourier mode ((k_x,k_z)) as

    [ \hat{\mathbf{u}}(k_x, y, k_z) = \sum_{n=0}^{N_y-1} \tilde{\mathbf{u}}_{n}(k_x,k_z)\, T_n(\eta), ]

    where (T_n) is the n-th Chebyshev polynomial.

  3. The Chebyshev coefficients (\tilde{\mathbf{u}}_{n}) are obtained with a Fast Chebyshev Transform (FCT), which is analogous to an FFT but uses the cosine recurrence relation.

  4. Define the 3‑D spectral energy density

    [ E(k_x, k_z, n) = \frac12 \bigl| \tilde{\mathbf{u}}_{n}(k_x,k_z) \bigr|^{2}. ]

  5. If you wish to obtain a scalar wavenumber magnitude you can construct a generalised wavenumber in the wall‑normal direction:

    [ k_y^{(n)} \equiv \frac{n\pi}{\delta}\quad\text{(approximate mapping of Chebyshev index to physical wavenumber)}. ]

    Then the pseudo‑3‑D isotropic spectrum is built by binning over shells in ((k_x, k_y^{(n)}, k_z)):

    [ E(k) = \sum_{\sqrt{k_x^{2}+k_y^{(n)2}+k_z^{2}} \in [k,\,k+\Delta k]} E(k_x,k_z,n). ]

    Caveat: Because the wall‑normal direction is not truly homogeneous, the interpretation of (k_y) is only approximate; nevertheless, the procedure is useful for visualising how energy is distributed across wall‑normal “scales”.

2.3.2 Practical algorithm (pseudo‑code)

# data: u[x, y, z]  (x,z periodic, y bounded)
# 1. FFT in x and z
U_kxkz = fft2(u, axes=(0,2))               # shape (Nx, Ny, Nz)

# 2. Chebyshev transform in y for each (kx,kz)
U_hat = np.zeros_like(U_kxkz, dtype=complex)
for ix in range(Nx):
    for iz in range(Nz):
        U_hat[ix,:,iz] = cheb_transform(U_kxkz[ix,:,iz])   # returns N_y coeffs

# 3. Spectral energy
E = 0.5 * np.abs(U_hat)**2                     # shape (Nx, Ny, Nz)

# 4. (optional) map Chebyshev index to k_y and bin in spherical shells
k_y = np.pi * np.arange(Ny) / delta
k_mag = np.sqrt(kx_grid**2 + k_y[:,None,None]**2 + kz_grid**2)
E_shell = bin_into_shells(k_mag, E, dk)

2.4 If you do not need a full 3‑D spectrum

Most experimental and DNS analyses of wall‑bounded turbulence use one‑dimensional spectra defined on planar cuts:

Spectrum Definition Typical use    
Streamwise spectrum (E_{uu}(k_x; y)) (E_{uu}(k_x; y) = \frac12\sum_{k_z} \hat{u}(k_x,y,k_z) ^{2}) Energy distribution along the streamwise direction at a given (y).
Spanwise spectrum (E_{uu}(k_z; y)) analogous Checks isotropy in the homogeneous plane.    
Pre‑multiplied spectrum (k_x E_{uu}(k_x; y)) Highlights contribution per logarithmic band. Useful for identifying the “(k^{-1})” region in the near‑wall logarithmic layer.    
Wall‑normal “spectral density” (E_{uu}(y;k_y)) obtained via Chebyshev/DCT as described above. Gives a scale‑by‑scale picture of how energy varies with distance from the wall.    

These spectra are local in (y): each wall‑normal location is treated independently, which is permissible because the flow is inhomogeneous only in that direction.

2.5 Defining scales in the wall‑normal direction

In a boundary layer we use inner (viscous) scaling and outer (geometric) scaling:

Region Length scale Velocity scale Typical non‑dimensional coordinate
Viscous sublayer ((y^+ \lesssim 5)) Viscous length (\ell_\nu = \nu / u_\tau) Friction velocity (u_\tau) (y^+ = y u_\tau / \nu)
Buffer layer ((5 \lesssim y^+ \lesssim 30)) Same as above Same (y^+)
Logarithmic layer ((30 \lesssim y^+ \lesssim 0.15 \, Re_\tau)) Outer length (\delta) (boundary‑layer thickness) (u_\tau) (y^+ = y u_\tau/\nu) and (y/\delta)
Outer region ((y/\delta \sim O(1))) (\delta) Free‑stream velocity (U_\infty) (y/\delta)

These non‑dimensional variables replace a single “global” integral length scale

Original question: Kolmogorov/Energy spectrum for turbulent boundary layer on Physics Stack Exchange, licensed CC BY-SA.