Analytical evaluation of techniques for use of a laser Doppler velocimeter to measure flow and turbulence*

C. Y. She; L. S. Wall

doi:10.1364/JOSA.65.000069

Journal of the Optical Society of America
Vol. 65,
Issue 1,
pp. 69-77
(1975)
•https://doi.org/10.1364/JOSA.65.000069

Analytical evaluation of techniques for use of a laser Doppler velocimeter to measure flow and turbulence

C. Y. She and L. S. Wall

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C. Y. She and L. S. Wall, "Analytical evaluation of techniques for use of a laser Doppler velocimeter to measure flow and turbulence*," J. Opt. Soc. Am. 65, 69-77 (1975)

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Abstract

Results of autocorrelation functions and power spectral densities of laser Doppler velocimeters in the reference-beam-heterodyning and the cross-beam-mixing mode are derived to handle turbulence measurements, Doppler ambiguity, spatial coherence, and effects of particle concentration and detection aperture. A useful definition of signal-to-noise ratio is given for both correlation and spectral analyses. The signal-to-noise calculation provides a direct comparison of the figures of merit between various arrangements of a laser Doppler velocimeter with methods for signal processing to measure optically the flow properties of a fluid under given experimental conditions.

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electric-field amplitude	uniform beam	gaussian beam
E₁(R)	$\frac{a}{R} {(\frac{P_{1}}{A_{0}})}^{1 / 2}$	$\frac{a}{R} {(\frac{P_{1}}{π σ^{2}})}^{1 / 2} exp [- \frac{x_{1}^{2} + z_{1}^{2}}{2 σ^{2}}]$
E₂(R)	$\frac{a}{R} {(\frac{P_{2}}{A_{0}})}^{1 / 2}$	$\frac{a}{R} {(\frac{P_{2}}{π σ^{2}})}^{1 / 2} exp [- \frac{x_{2}^{2} + z_{2}^{2}}{2 σ^{2}}]$
E_R(R)	$\frac{A_{0}}{λ R} {(\frac{P_{2}}{A_{0}})}^{1 / 2}$	$\frac{π σ^{2}}{λ R} {(\frac{P_{2}}{π σ^{2}})}^{1 / 2} exp [- \frac{x_{D}^{2} + z_{D}^{2}}{2 s^{2}}]$ ^a

magnitude	uniform beam	gaussian beam
I_R	$\frac{A_{0}}{λ^{2}} P_{2}$	$\frac{P_{2}}{Ω} erf (\sqrt π \frac{Ω}{Ω_{c}}) erf (\sqrt π \frac{Ω}{Ω_{c}})$
I_R₁	$\frac{a λ}{A_{0}} \sqrt (P_{1} P_{2})$	$\frac{a λ}{π σ^{2}} \sqrt (P_{1} P_{2}) exp [- (x_{1}^{2} + z_{1}^{2}) / 2 σ^{2}]$
I_R₂	$\frac{a λ}{A_{0}} P_{2}$	$\frac{a λ}{π σ^{2}} P_{2} exp [- (x_{2}^{2} + z_{2}^{2}) / 2 σ^{2}]$
I₁	$\frac{a^{2}}{A_{0}} P_{1}$	$\frac{a^{2}}{π σ^{2}} P_{1} exp [- (x_{1}^{2} + z_{1}^{2}) / 2 σ^{2}]$
I₂	$\frac{a^{2}}{A_{0}} P_{2}$	$\frac{a^{2}}{π σ^{2}} P_{2} exp [- (x_{2}^{2} + z_{2}^{2}) / 2 σ^{2}]$
I₁₂	$\frac{a_{1} a_{2}}{A_{0}} \sqrt (P_{1} P_{2})$	$\begin{array}{r} \frac{a_{1} a_{2}}{π σ^{2}} \sqrt (P_{1} P_{2}) exp [- (x_{1}^{2} + z_{1}^{2}) / 2 σ^{2}] \\ \cdot exp [- (x_{2}^{2} + z_{2}^{2}) / 2 σ^{2}] \end{array}$
$I_{11}^{(2)}$	$\frac{a_{1} (1) a_{1} (2)}{A_{0}} P_{1}$	$\begin{array}{r} \frac{a_{1} (1) a_{2} (2)}{π σ^{2}} P_{1} exp [- (x_{1}^{2} (1) + z_{1}^{2} (1)) / 2 σ^{2}] \\ \cdot exp [- (x_{1}^{2} (1) + z_{1}^{2} (1)) / 2 σ^{2}] \end{array}$
$I_{22}^{(2)}$	$\frac{a_{2} (1) a_{2} (2)}{A_{0}} P_{2}$	$\begin{array}{r} \frac{a_{1} (1) a_{2} (2)}{π σ^{2}} P_{2} exp [- (x_{2}^{2} (1) + z_{2}^{2} (1)) / 2 σ^{2}] \\ \cdot exp [- (x_{2}^{2} (1) + z_{2}^{2} (1)) / 2 σ^{2}] \end{array}$
$I_{12}^{(2)}$	$\frac{a_{1} (1) a_{2} (2)}{A_{0}} \sqrt (P_{1} P_{2})$	$\begin{array}{r} \frac{a_{1} (1) a_{2} (2)}{A_{0}} \sqrt (P_{1} P_{2}) exp [- (x_{1}^{2} (1) + z_{1}^{2} (1)) / 2 σ^{2}] \\ exp [- (x_{2}^{2} (1) + z_{2}^{2} (1)) / 2 σ^{2}] \end{array}$
Ω	$\frac{(2 x_{0}) (2 z_{0})}{R^{2}}$	$\frac{(2 x_{0}) (2 z_{0})}{R^{2}}$
Ω_c	$\frac{λ^{2}}{A_{0}}$	$\frac{λ^{2}}{π σ^{2}}$

	uniform beam	gaussian beam
g₁(τ)	q_b[X₁(τ) + Δx₁(τ)]q_b[Z₁(τ) + Δz₁(τ)]	exp[−X₁(τ) + Δx₁(τ))²/4σ²]exp[−Z₁(τ) + Δz₁(τ))²/4σ²]
$g_{1}^{'} (τ)$	q_b[X₁(τ) + Δx₁(τ)]q_b[Z₁(τ) + Δz₁(τ)]	exp[−X₁(τ) + Δx₁(τ))²/2σ²]exp[−Z₁(τ) + Δz₁(τ))²/2σ²]
N₁	ρ (b²l)	ρ (πσ²l)
$N_{1}^{'}$	ρ (b²l)	ρ (π/2)σ²l
$〈 g_{1}^{'} (τ) 〉$	f₁[b, X₁(τ)]f₁[b, Z₁(τ)]	f₂[σ, X₁(τ)]f₂[σ, Z₁(τ)]
〈g₁(τ)cos[┄]〉	$\begin{array}{l} f_{2} [b, (\vec{K} \cdot {\hat{x}}_{1}), X_{1} (τ)] f_{3} [b, (\vec{K} \cdot {\hat{z}}_{1}), Z_{1} (τ)] \\ \cdot exp [- {(\vec{K} \cdot {\hat{y}}_{1})}^{2} ξ^{2} (τ) / 2] \end{array}$	$\begin{array}{l} f_{4} [σ, (\vec{K} \cdot {\hat{x}}_{1}), X_{1} (τ)] f_{4} [σ, (\vec{K} \cdot {\hat{z}}_{1}), Z_{1} (τ)] \\ \cdot exp [- {(\vec{K} \cdot {\hat{y}}_{1})}^{2} ξ^{2} (τ) / 2] \end{array}$
where
$\begin{array}{l} f_{1} [b, X (τ)] = {[\sqrt (2 π) ξ (τ)]}^{- 1} exp [- X^{2} (τ) / 2 ξ^{2} (τ)] * q_{b} [X (τ)], \\ f_{2} [σ, X (τ)] = σ {[σ^{2} + ξ^{2} (τ)]}^{- 1 / 2} exp [- X^{2} (τ) / 2 (σ^{2} + ξ^{2} (τ))], \\ f_{3} [σ, (\vec{K} \cdot \hat{x}), X (τ)] = exp [- {(\vec{K} \cdot \hat{x})}^{2} ξ^{2} (τ) / 2] * {b {[\frac{sin {(\vec{K} \cdot \hat{x}) (b / 2)}}{(\vec{K} \cdot \hat{x}) (b / 2)}]}^{2} cos [(\vec{K} \cdot \hat{x}) X (τ)]}, \\ \begin{array}{l} f_{4} [σ, (\vec{K} \cdot \hat{x}), X (τ)] = & \sqrt{2} σ {[2 σ^{2} + ξ^{2} (τ)]}^{- 1 / 2} exp [- σ^{2} ξ^{2} (τ) {(\vec{K} \cdot \hat{x})}^{2} / (2 σ^{2} + ξ^{2} (τ))] \\ \cdot cos [(\vec{K} \cdot \hat{x}) X (τ) ξ^{2} (τ) / (2 σ^{2} + ξ^{2} (τ))] \end{array} \end{array}$
$\begin{array}{l} \vec{K} = {\vec{q}}_{1} - {\vec{q}}_{2}, & q_{b} (X) = \end{array}$

	uniform beam		gaussian beam
N₂	ρ (b²l)		ρ (πσ²l)
$N_{2}^{'}$	ρ (b²l)		ρ (π/2)σ²l
$g_{2}^{'} (τ)$	q_b[X₂(τ) + Δx₂(τ)]q_b[Z₂(τ) + Δz₂(τ)]		exp[−X₂(τ) + Δx₂(τ))²/2σ²]exp[−Z₂(τ) + Δz₂(τ))²/2σ²]
g(τ)	$q_{c_{x}} [X (τ) + Δ x (τ)] q_{c_{y}} [Y (τ) + Δ y (τ)]$		exp[−(X(τ) + Δx(τ))²/2(σ/cosθ)²]exp[−(Z(τ) + Δz(τ))²/2σ²]
	·q_b(Z(τ) + Δz(τ))		· exp(−(Y(τ) + Δy(τ))²/2(σ/sinθ)²]
N₁₂	ρ(c_xbc_y)		$ρ (\sqrt{π} σ / \sqrt{2} cos θ) (\sqrt{π} σ / \sqrt{2}) (\sqrt{π} σ / \sqrt{2} sin θ)$
$〈 g_{2}^{'} (τ) 〉$	f₁[b₁, X₂(τ)]f₁[b₁, Z₂(τ)]		f₂[σ, X₂(τ)]f₂[σ, Z₂(τ)]
〈g′(τ)〉	f₁[c_x, X(τ)]f₁[c_y, Y(τ)]f₁[b₁, Z(τ)]		f₂[(σ/cosθ), X(τ)]f₂[(σ/sinθ), Y(τ)]f₂[σ, Z(τ)]
〈g(τ)cos[┄]〉	$\begin{array}{l} f_{3} [c_{x}, (\vec{K} \cdot \hat{x}), X (τ)] f_{3} [c_{y}, (\vec{K} \cdot \hat{y}), Y (τ)] \\ \cdot f_{3} [b, (\vec{K} \cdot \hat{z}), Z (τ)] \end{array}$		$\begin{array}{l} f_{4} [(σ / \sqrt{2} cos θ), (\vec{K} \cdot \hat{x}), X (τ)] f_{4} [(σ / \sqrt{2}), (\vec{K} \cdot \hat{z}), Z (τ)] \\ \cdot f_{4} [(σ / \sqrt{2} sin θ), (\vec{K} \cdot \hat{y}), Y (τ)] \end{array}$
〈g₁₁(τ)cos[┄]〉	$\begin{array}{l} {f_{3} [b, ({\vec{K}}_{1} \cdot {\hat{x}}_{1}), X_{1} (τ)] f_{3} [b, ({\vec{K}}_{1} \cdot \hat{z}), Z_{1} (τ)] \\ {\cdot exp [- {({\vec{K}}_{1} \cdot {\hat{y}}_{1})}^{2} ξ^{2} (τ) / 2]}}^{2} \end{array}$		$\begin{array}{l} {f_{4} [σ, ({\vec{K}}_{1} \cdot {\hat{x}}_{1}), X_{1} (τ)] f_{4} [σ, ({\vec{K}}_{1} \cdot {\hat{z}}_{1}), Z_{1} (τ)] \\ {\cdot exp [- {({\vec{K}}_{1} \cdot {\hat{y}}_{1})}^{2} ξ^{2} (τ) / 2]}}^{2} \end{array}$
〈g₂₂(τ)cos[┄]〉	$\begin{array}{l} {f_{3} [b, ({\vec{K}}_{2} \cdot {\hat{x}}_{2}), X_{1} (τ)] f_{3} [b, ({\vec{K}}_{2} \cdot {\hat{z}}_{2}), Z_{2} (τ)] \\ {\cdot exp [- {({\vec{K}}_{2} \cdot {\hat{y}}_{2})}^{2} ξ^{2} (τ) / 2]}}^{2} \end{array}$		$\begin{array}{l} {f_{4} [σ, ({\vec{K}}_{2} \cdot {\hat{x}}_{2}), X_{2} (τ)] f_{4} [σ, ({\vec{K}}_{2} \cdot {\hat{z}}_{2}), Z_{2} (τ)] \\ {\cdot exp [- {({\vec{K}}_{2} \cdot {\hat{y}}_{2})}^{2} ξ^{2} (τ) / 2]}}^{2} \end{array}$
〈g₁₂(τ)cos[┄]〉	$\begin{array}{l} f_{3} [b, ({\vec{K}}_{1} \cdot {\hat{x}}_{1}), X_{1} (τ)] f_{3} [σ, ({\vec{K}}_{1} \cdot {\hat{z}}_{1}), Z_{1} (τ)] \\ \cdot exp [- {({\vec{K}}_{1} \cdot {\hat{y}}_{1})}^{2} ξ^{2} (τ) / 2] f_{3} [b, ({\vec{K}}_{2} \cdot {\hat{x}}_{2}), X_{2} (τ)] \\ \cdot f_{3} [b, ({\vec{K}}_{2} \cdot {\hat{z}}_{2}), Z_{2} (τ)] exp [- {({\vec{K}}_{2} \cdot {\hat{y}}_{2})}^{2} ξ^{2} (τ) / 2] \end{array}$		$\begin{array}{l} f_{4} [σ, ({\vec{K}}_{1} \cdot {\hat{x}}_{1}), X_{1} (τ)] f_{4} [σ, ({\vec{K}}_{1} \cdot {\hat{z}}_{1}), Z_{1} (τ)] \\ \cdot exp [- {({\vec{K}}_{1} \cdot {\hat{y}}_{1})}^{2} ξ^{2} (τ) / 2] f_{4} [σ, ({\vec{K}}_{2} \cdot {\hat{x}}_{2}), X_{2} (τ)] \\ \cdot f_{4} [σ, ({\vec{K}}_{2} \cdot {\hat{z}}_{2}), Z_{2} (τ)] exp [- {({\vec{K}}_{2} \cdot {\hat{y}}_{2})}^{2} ξ^{2} (τ) / 2] \end{array}$
and		$N_{12}^{'} ≃ N_{12}$ , g^′(τ) = g(τ), g₁₁(τ) = g₁(τ₁)g₁(τ₂), g₂₂(τ) = g₂(τ₁)g₂(τ₂), and g₁₂(τ) = g₁(τ₁)g₂(τ₂).
where		${\vec{K}}_{1} = {\vec{k}}_{1} - {\vec{k}}_{D}, {\vec{K}}_{2} = {\vec{k}}_{2} - {\vec{k}}_{D}$ , X₁ = X cosθ + Y sinθ, Y₁ = −X sinθ + Y cosθ,
		X₂ = X cosθ − Y sinθ, Y₂ = X sinθ + Y cosθ, Z₁ = Z₂ = Z, and τ₁ and τ₂
		denote particles 1 and 2, respectively.

S/N	reference beam	cross beam
R(τ)	$2 {(\frac{a}{λ})}^{2} (\frac{P_{1}}{P_{2}}) Ω_{c}^{2} {(\frac{κ_{R} Ω}{κ})}^{2} N / {[erf (\sqrt{π} \frac{Ω}{Ω_{c}})]}^{4}$	$\frac{1}{2 (N - 1)} + {(\frac{κ^{(2)}}{κ})}^{2}$
S(ω)^a	$η {(\frac{a}{λ})}^{2} (\frac{P_{1}}{ћ ω} τ_{c}) Ω_{c}^{2} (\frac{κ_{R} Ω}{κ}) N / {[erf (\sqrt{π} \frac{Ω}{Ω_{c}})]}^{2}$	$η {(\frac{a}{λ})}^{2} (\frac{P_{1}}{ћ ω} τ_{c}) Ω_{c}^{2}$
		$\cdot \frac{Ω}{Ω_{c}} [\frac{1}{2 {N (N - 1)}^{1 / 2}} + {(\frac{κ^{(2)}}{κ})}^{2} {N (N - 1)}^{1 / 2}]$

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Equations (38)

Journal of the Optical Society of America