Abstract
A theory of the effect of turbulence on the response of static pressure probes is given. To apply it to a given probe, the response in a laminar stream must be known as a function of orientation. Results are reported for Prandtl probes. For these the effect of turbulence is negligible at the levels encountered in pipe flows, but is highly significant at those prevailing in jets and other strongly turbulent systems. Experimental verification is given and the effect of turbulence length scale is demonstrated.
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Abbreviations
- (−):
-
time-mean value
- (^):
-
r.m.s. value (square root of the mean-square value; e.g., \(\hat u_x = {\text{(}}\overline {u_x^2 } {\text{)}}^{{\text{1/2}}} \)
- C(ϖ, ϱ):
-
impact pressure coefficient of probe
- Ce e :
-
C γ or C u
- C u :
-
two-point correlation coefficient between streamwise velocity fluctuations
- C γ :
-
two-point correlation coefficient between concentration fluctuations
- D :
-
Prandtl probe diameter, m
- D 0 :
-
jet nozzle diameter, m
- d :
-
pressure tap diameter, m
- E x :
-
probe response function defined by (17)
- e :
-
γ or u x , in general definitions of turbulence characteristics
- f x , f y , f z :
-
probability density functions of velocity fluctuations u x , u y and u z
- f xyz :
-
joint probability density function of velocity fluctuations u x , u y and u z
- G :
-
function defined by (16)
- g :
-
function defined by (22)
- h :
-
integral defined by (25)
- I :
-
function of h defined by (26)
- K :
-
coefficient defined by (21)
- L :
-
distance from pressure tap to probe tip, m Ppressure, Pa
- P b :
-
pressure at wall in boundary layer flows or at edge of turbulent flow in free turbulent flows, Pa
- P s :
-
pressure sensed by probe, Pa
- r :
-
radial coordinate, m
- Re D :
-
probe Reynolds number (U x D/v in a turbulent flow, x in the streamwise direction; UD/v in a laminar flow)
- Re D :
-
= U 0 D 0 /v, jet source (nozzle) Reynolds number
- U :
-
stream speed, m/s
- U :
-
stream velocity, m/s
- U i :
-
velocity component (i = x, y or z), m/s
- u i :
-
velocity fluctuation component (i = x, y or z), m/s
- \(\overline {u_i u_j } \) :
-
covariance of velocity fluctuations (i or j = x, y or z), m2/s2
- U n :
-
velocity component transverse to the probe axis, m/s (with U x parallel to the axis, U n 2 = U 2 y+U z 2)
- U 0 :
-
stream speed leaving jet nozzle, m/s
- U x , U y , U z :
-
velocity components in cartesian coordinates, m/s
- U x , U r , U ϱ :
-
velocity components in cylindrical coordinates, m/s
- u x , u y , u z :
-
components of velocity fluctuation in cartesian coordinates, m/s
- u x , u r u ϱ :
-
components of velocity fluctuation in cylindrical coordinates, m/s
- ν:
-
= Û n/Ū x, normalized r.m.s. cross-stream velocity (in a turbulent flow with x taken in the direction of the mean velocity vectorU, U 2 n = u 2 y + u 2 z )
- ν0 :
-
“standard” experimental value of ν, based on the best available measurements of velocity fluctuations
- ν* :
-
value of ν inferred by the present theory from measurements with a Prandtl static pressure probe
- w :
-
pressure tap depth, m
- W 1, W 2 :
-
half bandwidths defined by (24)
- x, y, z :
-
cartesian coordinates, m
- x, r, ϱ :
-
cylindrical coordinates, (m, m, rad)
- r, ϖ, ϱ:
-
spherical coordinates, (m, rad, rad)
- β:
-
= \(\overline {u_x^2 /} \overline {U_n^2 } \), ratio of streamwise and cross-stream mean-square velocity fluctuation levels
- Λ:
-
concentration of jet source (nozzle) fluid, kg/m3
- γ:
-
= Г − \(\bar \Gamma \), concentration fluctuation, kg/m3
- ϱ:
-
coaltitudinal angle (spherical coordinates), rad
- ϱ c :
-
integration limit in (26), rad
- Γ:
-
any integral turbulence length scale, m
- Γ u :
-
streamwise (“longitudinal”) integral length scale of streamwise velocity fluctuations, m
- Γ e :
-
Γγ or Γ u , m
- Γγ :
-
streamwise (“longitudinal”) integral length scale of concentration fluctuations, m
- γ:
-
= L/D, probe geometric characteristic
- v :
-
kinematic viscosity, m2/s
- ξ:
-
streamwise separation distance, m
- ϱ:
-
density, kg/m3
- ϖ:
-
longitudinal angle (spherical coordinates), rad
- Ψ:
-
= U/U x
- Ψ m :
-
value of Ψ defined by (23)
- Ψ1, Ψ2 :
-
integration limits in (25)
References
Batchelor, G. K. 1967: An introduction to fluid mechanics. Cambridge: Cambridge University Press
Becker, H. A.; Brown, A. P. G. 1969: Velocity fluctuations in turbulent jets and flames. Twelfth Symp. (Intern.) on Combustion, pp. 1059–1068. Pittsburgh: The Combustion Institute
Becker, H. A; Brown, A. P. G. 1974: Response of Pitot probes in turbulent streams. J. Fluid Mech. 62, 85–114
Becker, H. A. 1977: Mixing, concentration fluctuations, and marker nephelometry. Studies in Convection, Vol. 2 (B. E., Launder, Ed.), pp. 45–139. London, New York, San Francisco: Academic Press
Becker, H. A.; Hottel, H. C.; Williams, G. C. 1967: On the light scatter technique for the study of turbulence and mixing. J. Fluid Mech. 30, 259–284
Grandmaison, E. W.; Rathgeber, D. E.; Becker, H. A. 1982: Some characteristics of concentration fluctuations in free turbulent jets. Can. J. Chem. Eng. 60, 212–219
Miller, D. R.; Comings, E. W. 1967: Static pressure distribution in the free turbulent jet. J. Fluid Mech. 2, 1–16
Schlichting, H. 1959: Boundary Layer Theory. New York: McGraw-Hill
Shaw, R. 1960: The influence of hole dimensions on static pressure measurements. J. Fluid Mech. 1, 550–564
Townsend, A. A. 1956: The structure of turbulent shear flow. Cambridge: Cambridge University Press
Wygnanski, I.; Fiedler, H. 1969: Some measurements in the selfpreserving jet. J. Fluid Mech. 38, 577–612
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Cho, S.H., Becker, H.A. Response of static pressure probes in turbulent streams. Experiments in Fluids 3, 93–102 (1985). https://doi.org/10.1007/BF00276715
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DOI: https://doi.org/10.1007/BF00276715