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Flow visualization of compressible vortex structures using density gradient techniques

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Abstract

Mathematical results are derived for the schlieren and shadowgraph contrast variation due to the refraction of light rays passing through two-dimensional compressible vortices with viscous cores. Both standard and small-disturbance solutions are obtained. It is shown that schlieren and shadowgraph produce substantially different contrast profiles. Further, the shadowgraph contrast variation is shown to be very sensitive to the vortex velocity profile and is also dependent on the location of the peak peripheral velocity (viscous core radius). The computed results are compared to actual contrast measurements made for rotor tip vortices using the shadowgraph flow visualization technique. The work helps to clarify the relationships between the observed contrast and the structure of vortical structures in density gradient based flow visualization experiments.

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Abbreviations

a :

Unobstructed height of schlieren light source in cutoff plane, m

c :

Blade chord, m

f :

Focal length of schlieren focusing mirror, m

C T :

Rotor thrust coefficient, T/(ϱπΩ 2 R 4)

I :

Image screen illumination, Lm/m 2

l :

Distance from vortex to shadowgraph screen, m

n b :

Number of blades

p :

Pressure,N/m 2

p ∞ :

Ambient pressure, N/m 2

r, θ, z :

Cylindrical coordinate system

r c :

Vortex core radius, m

\(\bar r\) :

Non-dimensional radial coordinate, (r/r c )

R :

Rotor radius, m

νϑ:

Tangential velocity, m/s

γ :

Specific heat ratio of air

Γ ∞ :

Circulation (strength of vortex), m 2/s

ζ :

Non-dimensional quantity, Γ 2∞ ρ∞8π2γp ∞ r 2c

η :

Refractive index of fluid medium

η 0 :

Refractive index of fluid medium at reference conditions

κ :

Gladstone-Dale constant, m 3/kg

ϱ :

Density, kg/m 3

ϱ ∞ :

Density at ambient conditions, kg/m 3

\(\bar \rho\) :

Non-dimensional density, (ϱ/ϱ ∞)

σ :

Rotor solidity, (n b c/π R)

Ω :

Rotor rotational frequency, rad/s

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Authors and Affiliations

  1. Department of Aerospace Engineering, University of Maryland, 20742, College Park, MD, USA

    Ashish Bagai & J. Gordon Leishman

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  1. Ashish Bagai
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  2. J. Gordon Leishman
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Bagai, A., Leishman, J.G. Flow visualization of compressible vortex structures using density gradient techniques. Experiments in Fluids 15, 431–442 (1993). https://doi.org/10.1007/BF00191786

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