Abstract
The structure of fully-developed turbulence in a smooth pipe has been studied via wavenumber spectra for various friction velocities, namely, u τ,=0.61 and 1.2 m/s (the corresponding Reynolds numbers based on centerline velocity and pipe radius being respectively 134,000 and 268,000) at various distances from the wall, namely y + = 70, 200,400 and 1,000. For each distance from the wall, correlations of the longitudinal component of turbulence were obtained simultaneously in seven narrow frequency bands by using an automated data acquisition system which jointly varied the longitudinal (x) and transverse (z) separations of two hot-wire probes. The centre frequencies of the bandpass filters used correspond to a range of nondimensional frequencies ω + from 0.005 to 0.21. By taking Fourier transforms of these correlations, three-dimensional power spectral density functions and hence wavenumber spectra have been obtained at each y + with nondimensional frequency ω + and nondimensional longitudinal and transverse wavenumbers k +x and k +z as the independent variables. The data presented in this form show the distribution of turbulence intensity among waves of different size and inclination. The data reported here cover a wave size range of over 100, spanning a range of wave angles from 2° to 84°. The effects of friction velocity and Reynolds number on the distribution of waves, their lifetimes and convection velocities are also discussed.
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Abbreviations
- A :
-
wave strength function
- C x :
-
streamwise phase velocity
- C z :
-
circumferential phase velocity
- f :
-
wave intensity function
- k :
-
resultant wave number = [k 2x + k 2z ]1/2
- k x , k z :
-
longitudinal (x) and transverse (z) wavenumber respectively
- P(k + x , k +z , ω+):
-
power spectral density function in u
- R :
-
radius of pipe
- Re :
-
Reynolds number (based on centerline velocity and pipe radius)
- R uu (Δx +, Δz+, τ):
-
normalized correlation function in u
- R unu (Δx +, Δz+,¦ω+¦):
-
normalized filtered correlation function in u, as defined in equation (1)
- t :
-
time
- U :
-
mean velocity in the x-direction
- u, v, w :
-
turbulent velocities in the cartesian x, y and z directions respectively
- û, v, ŵ :
-
turbulent velocities in the wave coordinate x, ŷ and ž directions respectively
- u τ :
-
friction velocity
- x, y, z :
-
cartesian coordinates in the longitudinal (along the pipe axis), normal (to the pipe wall) and transverse (along the circumference of the pipe) directions respectively, as defined in Fig. 1
- α :
-
wave angle
- Δ :
-
difference between two quantities
- v :
-
kinematic viscosity
- τ :
-
time delay
- ω :
-
circular frequency (radians/s)
- +:
-
quantity nondimensionalized using u τand v
- overbar:
-
time average
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Lai, J.C.S., Bullock, K.J. & Hollis, P.G. Turbulence wavenumber spectra in fully-developed smooth pipe flow. Experiments in Fluids 12, 369–376 (1992). https://doi.org/10.1007/BF00193883
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DOI: https://doi.org/10.1007/BF00193883