What are premultiplied energy spectra?

One of the reason for using premultiplied spectra is meant for the $k_{-1}$ plateau as former author mentioned.

The other important reason is to show exactly the contribution to an entire spectrum of each wave number in a log-coordinate. Because the spectra are plotted mostly in log-coordinates, if we would like the area that the curve encloses to represent the turbulent kinetic energy, we need to multiply the spectra by the wave number.

Take the one-dimension streamwise spectrum for example, the integral of the 1D spectrum is equal to the streamwise component of TKE:

$$int_0^{+infty}Phi_{uu}(k_x) dk_x = overline{u^2}$$

But in the log-coordinates the area element is $d(ln(k_x))$ instead of $dk_x$ so we multiply the $Phi_{uu}$ by $k_x$ to obtain that :

$$int_0^{+infty} k_xPhi_{uu}(k_x) d(ln(k_x)) = int_0^{+infty} k_xPhi_{uu}(k_x) frac{1}{k_x} dk_x = overline{u^2}$$

So that you can find the area beneath the curve $k_xPhi_{uu}$ is meant for the turbulent kinetic energy contributed by the wave number $k_x$, in a log-coordinate.

As the previous answers point out, there's the log-log scale justification and there is also the Kolmogorov plateau. But I wanted to add that you REALLY need to be careful when using pre-multiplication to analyze your own data. Let's go through a simple example so you can see how much of a trap it can be:

Take an infinite bandwidth white noise signal. It has a flat spectrum (black line below). If you pre-multiply it and plot it in a linear-linear scale, it will have a ramp shape - as expected (blue line below).

Ok. Now let's low-pass this signal with a 2nd order butterworth filter with a cut-off frequency of 3 (arbitrary) and plot both the regular spectrum of this noise signal (black) and pre-multiplied spectrum (blue) in a linear-linear scale:

We know this noise signal is pretty uninteresting, and in fluid mechanics, this would simply indicate a regular turbulent cascade. However, a peak is produced in the pre-multiplied plot. That peak has little relevance, but it is approximately where the cutoff frequency is. For the sake of completeness, let's plot the same low-passed white noise signal in a semilog-x scale, which would be the reason why you would pre-multiply the spectrum to begin with (see Jinyan's answer for the mathematical discussion).

As you can see, there's a peak that is not at the cut-off frequency (fc=3), but is approximately where the cutoff is. The peak will appear at different frequencies depending on the order of the filter used (i.e., how fast it's rolling off).

With this in mind, we can safely say that pre-multiplied spectra will naturally produce a peak from a low-passed white noise signal. But signals found in fluid mechanics are naturally low-passed, white-noise-like. Therefore, the peaks are not meaningful. I agree with the original poster. There's a significant amount of papers out there identifying these peaks and analyzing them, looking for relevant "flow physics" (I work with shock-boundary layer interactions). It's, in my opinion, a huge fallacy. But it makes a boring story into an interesting one. That's probably why some academics fall for it.

See Matlab code below used to generate the plots in this discussion.

clear; clc; close all;

%Generates noise
g=randn(1e7,1);
n=10000; %Number of windows for PSD averaging
fs=20; %artificial sampling frequency for plotting

%%
[b,a]=butter(2,0.3); %Filters the noise
gf=filter(b,a,g);

%%
%Takes the power spectrum
[pw,w]=pwelch(gf,hann(length(g)/n),0,length(g)/n); %
[pwG,w]=pwelch(g,hann(length(g)/n),0,length(g)/n);
f=w*fs/(2*pi); %Converts from radian frequency to "physical frequency"

%%
%Plots
figure('COLOR','W','POSITION',[363 558 877 305]);
subplot(1,2,1)
plot(f,pw,'k-', 'linewidth',2);hold on;
plot(f,f.*pw,'b-', 'linewidth',2);
xlabel('f');
legend('P_{xx}', 'f*P_{xx}');
subplot(1,2,2)
semilogx(f,pw,'k-', 'linewidth',2);hold on;
semilogx(f,f.*pw,'b-', 'linewidth',2);
xlabel('f');
legend('P_{xx}', 'f*P_{xx}');

%%
figure('COLOR','W','POSITION',[363 558 877 305]);
subplot(1,2,1)
plot(f(2:end-1),pwG(2:end-1),'k-', 'linewidth',2);hold on;
plot(f(2:end-1),f(2:end-1).*pwG(2:end-1),'b-', 'linewidth',2);
xlabel('f');
legend('P_{xx}', 'f*P_{xx}');
subplot(1,2,2)
semilogx(f(2:end-1),pwG(2:end-1),'k-', 'linewidth',2);hold on;
semilogx(f(2:end-1),f(2:end-1).*pwG(2:end-1),'b-', 'linewidth',2);
xlabel('f');
legend('P_{xx}', 'f*P_{xx}');