The wavenumber spectra of aero-optical phase distortions by weakly compressible turbulence

Edwin Mathews; Kan Wang; Meng Wang; Eric J. Jumper

doi:10.1364/OE.27.005670

1. Introduction

As airborne optical systems have become more desirable for point-to-point aircraft communication and directed energy applications, understanding the physical mechanisms of optical beam distortions is essential to their successful implementation. While airborne beam transmission systems have been sought after and under various stages of development since the 1970’s, in modern systems the wavelength of the optical beam has decreased in order to increase the diffraction-limited intensity of the beam on a distant target. An undesirable consequence of smaller wavelength beams is an increased sensitivity to aero-optical distortions, or optical distortions due to density fluctuations induced by turbulent flow near the exit pupil of a beam director. Aero-optical effects can result in a severe degradation of the signal fidelity and on-target intensity of an optical beam as it propagates away from an aircraft [1].

Although atmospheric turbulence between the aircraft and its target is also an issue in airborne beam transmission, its effects on electromagnetic waves have been extensively studied [2–4] and adaptive-optics systems have been used to correct for its distortion for decades [5]. In comparison, aero-optical distortions are more difficult to correct using adaptive-optics methods because of much smaller characteristic time and length scales. Corrective systems for airborne lasers must be able to operate at frequencies on the order of kilohertz to mitigate aero-optical distortions compared to the hertz range required for atmospheric distortion corrections in ground-based telescopes. In addition, in studies of optical distortions caused by the atmosphere, a key and seemingly ubiquitous assumption is that local changes in the index of refraction are dominated by fluctuations in temperature (and humidity in some applications) and that the effects of pressure changes are negligible, which appears to be valid for the atmosphere [6]. In aero-optical flows, however, the effects of aerodynamics and compressibility enhance turbulent pressure fluctuations and it is inappropriate to neglect them [7].

The high sampling rate needed to probe aero-optical distortions constrained researchers to measuring time-mean statistics until the late 1980’s, when the first instrument described by Malley et al. [8] was developed. While these first devices were limited to resolving a single, small-aperture beam, further study of aero-optics was enabled by the development of high speed, two-dimensional Shack-Hartmann sensors [9,10]. With these new measurement tools and the advent of high-fidelity numerical simulations of turbulent flows, the modern era of aero-optics has emerged in which spatially and temporally resolved optical distortions are studied in relation to coherent turbulence structures and mitigation strategies are developed for application in airborne optical transmission systems. Reviews of modern aero-optical theory, measurement, computational techniques, and applications can be found in the articles by Jumper & Fitzgerald [1], Wang et al. [11], and Jumper & Gordeyev [10].

Time-resolved measurement techniques have allowed the determination of the power spectral density of aero-optical distortions caused by a variety of flows including subsonic [12–14] and supersonic [15] boundary layers, free shear layers and jets [16, 17], and bluff body flows [18]. However, not much analysis has been done to explain the spectral behavior within a well defined theoretical framework. Exceptions are the works by Gao et al. [19] and Prasad [20, 21]. Gao et al. analyzed the universal behavior of the phase distortion structure functions in optical transmissions through a supersonic boundary layer, and Prasad developed statistical models for optical beam propagation through aero-optical fields accounting for the effects of pressure dominant index-of-refraction fluctuations and large-scale turbulence anisotropy. Each of these works examined the behavior of aero-optical phase distortion spectra or structure functions, but none investigated with depth changes in the phase distortion spectra due to differences in the source of index-of-refraction fluctuations.

Although there is a body of literature on the propagation of electromagnetic waves through the atmosphere, the results are not directly applicable to aero-optical effects. It is desirable to extend well-known theories of atmospheric optical propagation to the study of beam propagation through an aero-optical field. The goal of the current work is to derive a general expression for turbulent index-of-refraction fluctuations and then apply wave propagation theory to describe the spatial spectral behavior of aero-optical distortions, particularly when turbulent pressure fluctuations may play a significant role in the refractive index. First, an approximation of the index-of-refraction spectrum will be constructed starting from the ideal gas law, followed by its application to find an approximation for the spectrum of phase distortions for a beam propagating through an aero-optical field. Results of high-fidelity flow simulations for a weakly compressible shear layer will then be presented to verify theoretical results.

2. Theoretical description of aero-optical phase fluctuations

Because refractive index fluctuations in aero-optical applications are weak and the region where aero-optical distortions occur is typically small [11], simplifying assumptions valid for aero-optics allow for a reduction of Maxwell’s equations to an expression for the scalar component of the electric field, 𝒰, at an optical wavenumber, k,

𝒰 (x, y, L) = 𝒰 (x, y, 0) exp [- i k \int_{0}^{L} n (x, y, z) d z],

where n is the index of refraction, z is the coordinate in the direction of beam propagation, and the aero-optically active turbulence region is in the range 0 ≤ z ≤ L. Equation (1) suggests that aero-optical effects only aberrate the phase of a beam and the amplitude, 𝒰(x, y, 0), is unchanged as a beam propagates through an aero-optically active flow region.

Since phase distortions are the principal aero-optical effect, theoretical research into aero-optics has historically focused on developing statistical descriptions of distortions in the optical phase term of Eq. (1), ϕ = k ∫ ndz = kOPL, where OPL is the optical path length. The optical phase distortion, ϕ′, represents the relative difference in the phase over an optical aperture and is proportional to the optical path difference, or OPD. The latter is the relative difference in OPL over an optical aperture. It can be shown that for a statistically homogeneous index-of-refraction field, ϕ′ = k ∫ n′dz = kOPD, where the fluctuating index of refraction, n′, is given by n′ = n − 〈n〉, and angle brackets denote spatial averaging. This relationship between ϕ′ and n′ suggests that to develop an expression for the phase-distortion spectrum for aero-optics, an expression for the fluctuating refractive-index spectrum must be derived first.

2.1. Refractive-index spectrum

The index of refraction for air can be related to its density via the Gladstone-Dale relation, n = 1 + K_GDρ, where ρ is the density of air and K_GD is the Gladstone-Dale coefficient, which can be assumed to be constant in the band of optical wavelengths of aero-optical interest. The density can be related to the temperature, T, and pressure, p, of air via the ideal gas law, ρ = p/RT, where R is the specific gas constant for air. Combining the two equations, an expression for the index of refraction of air can be formulated based on the temperature and pressure,

n = 1 + K_{GD} \frac{p}{RT} .

Any fluctuating quantity g can be cast into the sum of its time mean, ḡ, and fluctuating component, g′, as g = ḡ + g′. If g is taken to be statistically homogeneous in space and stationary in time then temporal and spatial averages are equivalent, g′ = g − 〈g〉 = g − ḡ. Applying this decomposition to variables in Eq. (2) and assuming that T′/T̄ is small and the flow field is statistically homogeneous, to first order the fluctuating component of index of refraction can be approximated as

n^{'} = K_{GD} ρ^{'} \approx K_{GD} \bar{ρ} (\frac{p^{'}}{\bar{p}} - \frac{T^{'}}{\bar{T}}) .

The three-dimensional autocorrelation function of the index of refraction is defined as

ℛ_{n n} (x_{0}, r) = \bar{n^{'} (x_{0}, t) n^{'} (x_{0} + r, t)},

where x₀ = x₀î + y₀ĵ + z₀k̂ is the point about which the correlation function is defined and r = r_xî+ r_yĵ + r_zk̂ is the displacement vector relative to x₀. Since the flow field is assumed to be homogeneous, ℛ_nn is independent of x₀ and, using Eq. (3), can be expanded as,

ℛ_{n n} (r) = K_{GD}^{2} ℛ_{ρ ρ} (r) \approx {(K_{GD} \bar{ρ})}^{2} (\frac{ℛ_{p p} (r)}{{\bar{p}}^{2}} + \frac{ℛ_{T T} (r)}{{\bar{T}}^{2}} - \frac{ℛ_{T p} (r)}{\bar{p} \bar{T}} - \frac{ℛ_{p T} (r)}{\bar{p} \bar{T}}) .

Here ℛ_Tp and ℛ_pT are the temperature-pressure and pressure-temperature cross-correlation functions, respectively. The cross-correlation function for two fluctuating quantities g′ and h′ are defined as

ℛ_{g h} (x_{0}, r) = \bar{g^{'} (x_{0}, t) h^{'} (x_{0} + r, t)} .

In a homogeneous field, ℛ_gh(r) = ℛ_hg(−r). The three-dimensional wavenumber spectrum of the refractive index can be obtained by a three-dimensional Fourier transform of Eq. (5),

Φ_{n n} (K) = K_{GD}^{2} Φ_{ρ ρ} (K) \approx {(K_{GD} \bar{ρ})}^{2} (\frac{Φ_{p p} (K)}{{\bar{p}}^{2}} + \frac{Φ_{T T} (K)}{{\bar{T}}^{2}} - \frac{2 Re [Φ_{p T} (K)]}{\bar{p} \bar{T}}),

where K = κ_xî + κ_yĵ + κ_zk̂ is the wavenumber vector.

This result for the refractive-index spectrum is very similar in form to expressions found by Friehe et al. [22], Hill et al. [6], and McBean & Elliott [23] in their investigations of index-of-refraction fluctuations in the atmosphere. In the first two of these investigations, however, the pressure contribution was appropriately assumed negligible in comparison to the optical effects of temperature and humidity. As a result, expressions were found for the index-of-refraction spectra as a function of fluctuating temperature spectra, fluctuating humidity spectra, and temperature-humidity cospectra. McBean & Elliott [23] derived an expression for the refractive-index spectrum using all three components (pressure, temperature, and humidity) but found by direct simultaneous measurements of the components that contributions from pressure fluctuations were small. In typical aero-optical applications, the effect of humidity is negligible but the similarity between Eq. (7) and expressions presented in these earlier works is noteworthy.

If the turbulent flow is assumed to be statistically both homogeneous and isotropic, the three-dimensional spectrum is related to the one-dimensional energy spectrum, E, via [24] E(κ) = 4πκ²Φ(κ), where κ = ‖K‖ is the scalar wavenumber. Equation (7) can then be rewritten in terms of the one-dimensional energy spectrum. If it is further assumed that the Reynolds number is large and the flow is only weakly compressible, velocity statistics can be approximated by their incompressible representations. To examine the inertial subrange behavior of index-of-refraction fluctuations, the pressure spectrum based on Kolmogorov’s scaling for the inertial subrange [25] is used to obtain,

\frac{E_{p p} (κ)}{{\bar{p}}^{2}} = \frac{B_{p} {\bar{ρ}}^{2} ε^{4 / 3} κ^{- 7 / 3}}{{\bar{p}}^{2}} = \frac{γ^{2}}{{\bar{a}}^{4}} B_{p} ε^{4 / 3} κ^{- 7 / 3},

where ā is the mean sound speed, γ is the ratio of specific heats for air, ε is the turbulence dissipation rate, and B_p is a constant. Similarly, an inertial subrange spectrum of temperature can be assumed based on the Kolmogorov-Obukhov-Corrsin prediction for the one-dimensional temperature spectrum [26,27],

\frac{E_{T T} (κ)}{{\bar{T}}^{2}} = \frac{β χ ε^{- 1 / 3} κ^{- 5 / 3}}{{\bar{T}}^{2}} = \frac{γ^{2}}{{\bar{a}}^{4}} R^{2} β χ ε^{- 1 / 3} κ^{- 5 / 3},

where β is a constant, χ is the average temperature dissipation rate given as χ = 2D_m〈∇T · ∇T〉, and D_m is the coefficient of molecular diffusion. Combining Eqs. (7)–(9) and the relation between the one-dimensional and three-dimensional turbulence spectrum leads to

Φ_{n n} (κ) \approx \frac{{(γ K_{GD} \bar{ρ})}^{2}}{4 π {\bar{a}}^{4}} (B_{p} ε^{4 / 3} κ^{- 13 / 3} + R^{2} β χ ε^{- 1 / 3} κ^{- 11 / 3} - \frac{2 R}{\bar{ρ} κ^{2}} Re [E_{p T} (κ)]) .

Clearly, Eq. (10) indicates that the spectral behavior of the index of refraction depends on contributions from the pressure, temperature, and the pressure-temperature cospectra. This expression, derived for the refractive-index spectrum in aero-optical turbulence, is more general than the classic expression used in most atmospheric turbulence studies,

Φ_{n n} (κ) = 0.033 C_{n}^{2} κ^{- 11 / 3},

where

C_{n}^{2}

is the refractive index structure coefficient that accounts for the strength of the turbulence and its value is proportional to the variance of the index of refraction. A comparison of the wavenumber dependence between Eqs. (10) and (11) shows that the spectrum for refractive index commonly used for atmospheric turbulence only accounts for the effect of temperature fluctuations. It can be expected that the expressions are equivalent when pressure fluctuations are negligible. By using the more general spectral representation derived here, we can proceed to develop theoretical expressions for the phase distortion by flows where pressure fluctuations are appreciable.

An implication of this result is that for some flow conditions, density and refractive-index spectra may be dominated by either temperature or pressure fluctuations and as such, can exhibit different spectral behaviors. This phenomenon was observed by Winarto & Davis in their studies of density fluctuations in turbulent jets [28]. When the static temperature of the jet differed from the ambient air temperature, the one-dimensional density spectrum had an inertial subrange with a slope −m_ρ = −5/3 indicating temperature dominant density fluctuations. When the static temperature of the jet was matched to the ambient, a spectral slope near −7/3 was observed, indicating that the turbulent density fluctuations were dominated by pressure fluctuations.

It is important to note that the one-dimensional spectral slopes of −m_T = −5/3 and −m_p = −7/3 mentioned above for the temperature and pressure spectra, respectively, are for homogeneous and isotropic turbulence at very high Reynolds numbers. Both experimentally and computationally, the spectral roll-off for temperature and pressure have been observed to be slower when the flow conditions are different and/or the Reynolds number is moderate [29–32].

2.2. Phase correlation of a plane wave propagating through an aero-optical field

To develop an expression for the spectral behavior of the phase distortion, ϕ′, we first construct the autocorrelation function for the phase distortion in the beam cross-section plane at a distance z from the aperture,

ℛ_{ϕ ϕ} (x_{0}, r_{⊥}, z) = \bar{ϕ^{'} (x_{0}, z) ϕ^{' *} (x_{0} + r_{⊥}, z)},

where r_⊥ = r_xî + r_yĵ, ϕ′^* is the complex conjugate of ϕ′, and to simplify notation, the same symbols (x,y,z) are used for the coordinate system local to the beam even though it may or may not be aligned with the flow coordinate system. Assuming that the beam is initially collimated, its phase distortion in the aero-optical region can be analyzed in terms of plane wave propagation through a variable refractive-index field. If the index-of-refraction field is assumed homogeneous and hence its wavenumber spectrum is radially symmetric, Ishimaru [4] shows that, after the beam passes through the turbulence region (0 ≤ z ≤ L),

ℛ_{ϕ ϕ} (r_{⊥}, L) = 2 π^{2} k^{2} L \int_{0}^{\infty} κ_{⊥} J_{0} (κ_{⊥} r_{⊥}) f_{ϕ^{'}} (κ_{⊥}) Φ_{n n} (κ_{⊥}) d κ_{⊥},

where J₀ is the zeroth order Bessel function of the first kind,

κ_{⊥}^{2} = κ_{x}^{2} + κ_{y}^{2}

,

r_{⊥}^{2} = r_{x}^{2} + r_{y}^{2}

, and f_ϕ′ is a spectral filter function,

f_{ϕ^{'}} (κ_{⊥}) = 1 + \frac{sin (κ_{⊥}^{2} L / k)}{κ_{⊥}^{2} L / k} .

Aero-optical flows are typically inhomogeneous in the propagation direction. As such, a formulation for the phase autocorrelation function based on an index-of-refraction spectrum that varies in the propagation direction must be determined. If the spectral dependence on the scalar transverse wavenumber, κ_⊥, is independent of the propagation direction, the index-of-refraction spectrum, Φ_nn can be put into a separable form Φ_nn = Φ_{1_n}(z)Φ_{2_n}(κ_⊥) [4], where Φ_{1_n}(z) is proportional to the variance of the index of refraction at location z along the beam path and Φ_{2_n}(κ_⊥) does not change along the beam path. With this decomposition of the refractive-index spectrum, Eq. (13) can be generalized as [4]

ℛ_{ϕ ϕ} (r_{⊥}, L) = {(2 π k)}^{2} \int_{0}^{L} Φ_{1_{n}} (z) [\int_{0}^{\infty} κ_{⊥} J_{0} (κ_{⊥} r_{⊥}) {cos}^{2} (\frac{L - z}{2 k} κ_{⊥}^{2}) Φ_{2_{n}} (κ_{⊥}) d κ_{⊥}] d z .

It is noted that the component of autocorrelation function within the brackets is similar to Eq. (13) with the only difference being the functions that act as filters on the refractive-index spectrum. The outer integral has no effect on the transverse dependence of the correlation function and only provides varying weights for regions of the flow field along the propagation path.

2.3. One-dimensional wavenumber spectrum of aero-optical distortions

In this section, the result for the phase distortion correlation will be used to derive expressions for the one-dimensional wavenumber spectrum of optical distortions in homogeneous turbulent flow fields, focusing on the spectral behavior in the inertial subrange. The one-dimensional spectrum of phase distortion along the x-direction, E_ϕϕ(κ_x), can be calculated by taking the Fourier transform of the phase autocorrelation in the r_⊥ = r_xî direction. Assuming radial symmetry in the correlation function as before, one obtains

E_{ϕ ϕ} (κ_{x}) = \int_{- \infty}^{\infty} ℛ_{ϕ ϕ} (r_{x}, L) e^{- i r_{x} κ_{x}} d r_{x} = 2 \int_{0}^{\infty} ℛ_{ϕ ϕ} (r_{x}, L) cos (r_{x} κ_{x}) d r_{x} .

In addition, the correlation for homogeneous turbulence given by Eq. (13) can be simplified by assuming that the spatial wavenumber in the inertial region is large such that

κ_{⊥}^{2} ≫ k / L

. This assumption leads to f_ϕ′ (κ_⊥) ≈ 1 and facilitates a study of the high-wavenumber optical distortions in locally homogeneous turbulence, as in Ishimaru [4]. Using this simplification for R_ϕϕ, Eq. (16) becomes

E_{ϕ ϕ} (κ_{x}) = 4 π^{2} k^{2} L \int_{0}^{\infty} [\int_{0}^{\infty} κ_{x}^{'} J_{0} (κ_{x}^{'} r_{x}) Φ_{n n} (κ_{x}^{'}) d κ_{x}^{'}] cos (r_{x} κ_{x}) d r_{x} .

Since the inertial subrange spectrum is not valid at low wavenumbers, it alone cannot be used to evaluate the integral. To evaluate the integral in Eq. (17), a spectral model used by Tatarski [3] and Ishimaru [4], which is a simple form of the von Karman model of the turbulent spectrum, is employed. In this model, the spectrum of a fluctuating quantity g′ is given by,

Φ_{g g} (κ) = C_{g g} \bar{g^{' 2}} ℓ^{3} {(1 + κ^{2} ℓ^{2})}^{- m_{g}^{3 D} / 2},

where ℓ is a characteristic length scale [33] and

- m_{g}^{3 D}

is the slope of the three-dimensional spectrum. This spectrum separates the inertial subrange from low-wavenumber scales of turbulence at a peak value of κℓ = 1, corresponding to a peak turbulence length scale equal to 2πℓ. The integral of the model spectrum across all wavenumbers is equal to

\bar{g^{' 2}}

, and the value of C_gg is a constant that satisfies this relation. The spectral level given by Eq. (18) tends to a constant as κℓ → 0 and decreases as

κ^{- m_{g}^{3 D}}

for κℓ ≫ 1 in the inertial subrange of the turbulence.

If this model spectrum is used for turbulent density fluctuations, a general expression for the wavenumber spectrum of the index of refraction can be found without using the linearized ideal gas law approximation. The dependence of the one-dimensional phase spectrum on density spectrum can then be evaluated from Eq. (17) to obtain

E_{ϕ ϕ} (κ_{x}) = 2 π^{5 / 2} k^{2} K_{GD}^{2} C_{ρ ρ} \bar{ρ^{' 2}} L ℓ^{2} {(1 + κ_{x}^{2} ℓ^{2})}^{\frac{1 - m_{ρ}^{3 D}}{2}} \frac{Γ ((m_{ρ}^{3 D} - 1) / 2)}{Γ (m_{ρ}^{3 D} / 2)} .

where

Γ (s) = \int_{0}^{\infty} x^{s - 1} exp (- x) d x

is the gamma function, and

- m_{ρ}^{3 D}

is the three-dimensional spectral slope of density. Importantly, Eq. (19) indicates that for κ_xℓ ≫ 1, if the three-dimensional density spectrum has a spectral slope of

- m_{ρ}^{3 D}

, the one-dimensional phase fluctuation spectrum will have a spectral slope of

- (m_{ρ}^{3 D} - 1)

. Equivalently, if the inertial subrange of the one-dimensional density spectrum has a spectral slope of −m_ρ, the one-dimensional phase fluctuation spectrum will have a slope of −(m_ρ + 1) because

m_{ρ} = m_{ρ}^{3 D} - 2

.

A similar relation can be found for the temperature, pressure, and pressure-temperature components in the linearized spectrum of index-of-refraction fluctuations derived earlier. The inertial-subrange model spectra used to derive Eq. (10) are replaced with the von Karman model of the turbulence spectrum, resulting in the following expression for the refractive-index spectrum:

\begin{array}{l} Φ_{n n} (κ) \approx \frac{{(γ K_{GD} \bar{ρ})}^{2}}{4 π {\bar{a}}^{4}} (B_{p} ε^{4 / 3} ℓ^{13 / 3} {[1 + κ^{2} ℓ^{2}]}^{- 13 / 6} \\ + R^{2} β χ ε^{- 1 / 3} ℓ^{- 11 / 3} {[1 + κ^{2} ℓ^{2}]}^{- 11 / 6} - \frac{2 R}{\bar{ρ} κ^{2}} Re [E_{p T} (κ)]) . \end{array}

Using this expression in Eq. (17) and integrating for E_ϕϕ in terms of the temperature spectrum, pressure spectrum, and cospectrum gives

\begin{matrix} E_{ϕ ϕ} (κ_{x}) \approx π {(\frac{γ K_{GD} \bar{ρ} k L^{1 / 2}}{{\bar{a}}^{2}})}^{2} (\frac{\sqrt{π}}{2} B_{p} ε^{4 / 3} ℓ^{10 / 3} {[1 + κ_{x}^{2} ℓ^{2}]}^{- 5 / 3} \frac{Γ (5 / 3)}{Γ (13 / 6)} \\ + \frac{\sqrt{π}}{2} R^{2} β χ ε^{- 1 / 3} ℓ^{8 / 3} {[1 + κ_{x}^{2} ℓ^{2}]}^{- 4 / 3} \frac{Γ (4 / 3)}{Γ (11 / 6)} \\ - \frac{8 π R}{\bar{ρ}} \int_{0}^{\infty} [\int_{0}^{\infty} J_{0} (κ_{x} r_{x}) Re [Φ_{p T} (κ_{x})] d κ_{x}] cos (r_{x} κ_{x}) d r_{x}) . \end{matrix}

Notably, the wavenumber spectrum contains two different spectral slopes as κ_x becomes large, −10/3 and −8/3, in addition to an unknown contribution from the pressure-temperature cospectrum. The predominant slope is dependent on the conditions of the flow, as seen in atmospheric flows where the −8/3 slope is almost exclusively observed due to the dominance of temperature fluctuations in the turbulent atmosphere.

The spectral slope of turbulence quantities and their dependence on Reynolds number and flow structure have direct consequences for the spectral slope of optical phase distortions. If the flow of interest is of low Reynolds number and/or in shear, it is possible for the slope of the one-dimensional temperature spectrum to take the value of −4/3 instead of −5/3 based on the Kolmogorov prediction [29]. Similarly, the pressure spectral slope for low Reynolds numbers can be −5/3 compared to the Kolmogorov value of −7/3 [30–32]. In this scenario it can be expected that the temperature will contribute to the one-dimensional phase spectrum a component with a spectral slope of −7/3 and the pressure will contribute a component of slope −8/3. The total phase spectral behavior will then depend on the strength of each of these contributions together with the contribution from the cospectra.

Equations (19) and (21) are derived based on the assumption that the flow is homogeneous in the propagation direction. As discussed earlier, for inhomogeneous flows it is sometimes possible to model the index-of-refraction spectrum in a separable form where its wavenumber dependence is separated from strength of the refractive index fluctuations along the integration path. This may be a reasonable assumption for certain flow configurations but may be inadequate for flows such as boundary layers where the spectral dependence on wavenumber can change with the distance from the wall. For these flows, a more complex coupling between the index of refraction magnitude and spectral dependence on wavenumber must be considered.

For the case where the decoupling of fluctuation magnitude and wavenumber dependence is appropriate and the refractive-index variance varies smoothly in z, Woo & Ishimaru [34] showed that the phase spectral slope has the same dependence on wavenumber as in the homogeneous case. Therefore, in the case where this decoupling is approximately valid, relations for the spectral slope of phase distortions are expected to be proportionally the same as those found here for the homogeneous turbulence case. This behavior can also be expected at high wavenumbers in general turbulent flows, where the large-scale structures may have preferential correlation directions but local isotropy is expected at high wavenumbers.

3. Numerical simulations of aero-optical distortions by turbulent shear layers

To verify the theoretical results presented in the preceding sections, three numerical simulations of weakly compressible, temporally evolving turbulent shear layers (also known as mixing layers) have been conducted. Temporally evolving shear layers are chosen due to its simple configuration, availability of previous simulation results for validation, and similarity to separated shear layers commonly found in aero-optical applications. While other fundamental flows could have potentially served the same purpose, no other flow contains the simplicity of the shear layer while allowing easy introduction of a density gradient that could be utilized to isolate the individual effects of temperature and pressure on turbulent density fluctuations.

3.1. Computational method

Large-eddy simulation (LES) is employed to compute the turbulent shear layers at a sufficiently high Reynolds number so that the inertial subrange can be clearly identified. An unstructured-mesh finite-volume code named CharLES developed by Cascade Technologies Inc. [35] is used to solve the spatially-filtered compressible Navier-Stokes equations in space and time. On uniform Cartesian meshes, as used in this work, CharLES uses a fourth-order central flux method for the Euler flux and a second-order scheme for the dissipative fluxes. The solver uses an explicit third-order Runge-Kutta method in time, and the Vreman subgrid-scale model [36,37] is employed to model the effects of sub-grid scale motions.

The domain size and simulation initialization are similar to those of the baseline case (termed A3) in the direct numerical simulation (DNS) of Pantano & Sarkar [38] at a much lower Reynolds number. In Fig. 1, the domain for the current numerical investigations is shown, where U₁ corresponds to the upper stream of the flow and U₂ the lower stream. ΔU = U₂ − U₁ = 0.6a₀(0), where a₀(0) is the speed of sound at the center of the shear layer at the simulation start, so that the freestream velocities at the top and bottom of the mixing layer are equal to ±0.3a₀(0). A common measure used to quantify compressibility effects in shear flows is the convective Mach number, M_c = ΔU/(a₁ + a₂), where a₁ is the sound speed in the upper stream of the flow and a₂ is the sound speed in the lower stream of the flow. In all simulation configurations, M_c = 0.3 so that the effect of fluid compressibility is weak and on the borderline of the typical incompressible flow approximation.

Fig. 1 Spanwise cross-section of the computational domain for the temporally evolving mixing layer simulations.

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The domain dimensions are defined relative to the initial momentum thickness of the mixing layer, δ_θ(0), and are equal to L_x = 345δ_θ(0) and L_y = 86δ_θ(0) in the streamwise and spanwise directions, respectively, to be the same as in the A3 case by Pantano & Sarkar [38]. It should be noted that in the current simulation, the streamwise coordinate is x, the spanwise coordinate is y, and the vertical coordinate is z. The coordinate system is chosen such that the aero-optical distortion is calculated along the z coordinate in keeping with previous notation. The computational domain is periodic in both the streamwise and spanwise directions. At the top and bottom of the domain are numerical sponge layers used to damp and prevent the reflection of acoustic disturbances in the finite domain [39]. The vertical size of the domain is L_z = 200δ_θ(0), approximately 15% larger than that in the A3 case of Pantano & Sarkar to accommodate the additional length of the sponge layer.

The flow is first initialized using a hyperbolic tangent profile for the streamwise velocity, u(0) = 0.5ΔU tanh [−z/(2δ_θ(0))], and zero for the other two velocity components. To promote the transition of the shear layer to a turbulent state, a random field of divergence-free velocity perturbations with a prescribed isotropic spectrum as used by Pantano & Sarkar, Φ_{u_iu_i} (κ) ∼ (κ/κ₀)⁴ exp [−2 (κ/κ₀)²], is applied to the initial velocity field. This is accomplished using the method described by Rogallo [40] and the peak wavenumber κ₀ is selected such that there are 24 peak wavelengths across the streamwise domain, κ₀L_x/2π = 24. The root-mean-square (rms) value of the fluctuations is set to be 0.1ΔU, and the fluctuations are confined to −50 < z/δ_θ(0) < 50 by applying an exponential ramping function.

The initial temperature also follows a hyperbolic tangent profile of the same form as the initial streamwise velocity. Since the pressure is held constant across the mixing layer, the initial density profile also has a hyperbolic tangent profile as required by the ideal gas law. Three simulation configurations with different ratios of lower to upper stream density, s = ρ₂/ρ₁, are used to investigate the effect of temperature and pressure on optical phase distortions. To investigate the flow condition where the pressure is expected to play the largest role in index-of-refraction fluctuations, s is set to 1.0. In the second case, s is equal to 1.05 to represent a more realistic subsonic separated shear layer. This value is found by examining the ratios of density across separated shear layers in previous computations; simulations of flow over a hemisphere-on-cylinder turret found s across the separated shear layer to be approximately 1.03 [41] and in computations involving cylindrical turrets [18], density ratios in the range of 1.06 − 1.075 were observed across the separated shear layer. To investigate conditions where temperature effects are expected to completely dominate index-of-refraction fluctuations, s = 1.35 is used. The initial freestream values of thermodynamic quantities for the three simulations can be found in Table 1.

Table 1. Initial freestream values for shear layer simulations^a

View Table

The number of control volumes in each direction of the computational domain is N_x ×N_y ×N_z = 1200 × 300 × 696, which provides 2.3 times the resolution of Pantano & Sarkar [38] relative to δ_θ(0) in each direction, and the mesh spacing is approximately equal in all three directions with a uniform distribution. To improve the convergence of statistical results, all statistical quantities presented have been ensemble averaged over three individual simulations for each of the three computational configurations along with spanwise and streamwise averaging when possible. The three ensemble simulations are identical except for different initial random perturbation fields.

In each simulation, the Reynolds number based on the momentum thickness, Re_θ = ΔUδ_θ/ν̄₀, where ν̄₀ is the average kinematic viscosity in the center-plane of the flow, is initially 1600. This is ten times the Reynolds number in the DNS of Pantano & Sarkar. The decision to increase the Reynolds number from the DNS value was made after initial calculations at the DNS Reynolds number did not exhibit a strong, identifiable inertial subrange.

3.2. Flow field results

Major qualitative differences among the three shear layer configurations can be seen in Fig. 2 where contours of density normalized by the initial center-plane density, ρ₀, are shown for a section of the shear layer at a time instant after the flow has fully transitioned to turbulence. Three different physical mechanisms of density fluctuations are observed from the figures. The first mechanism, most easily identified in Figs. 2(a) and 2(b), is due to the drop in pressure in the cores of vortex structures. The second is a decrease in density due to viscous heating and is only clearly visible in the s = 1.0 shear layer shown in Fig. 2(a). The third mechanism is the turbulent mixing of different density air streams. This may be the only density fluctuation mechanism visible in Fig. 2(c) and is partially visible in Fig. 2(b). While the presence of large scale vortical structures is not as visible in the density field from the s = 1.35 simulation, periodic ejections of high and low density fluid into the opposite freestream can be seen and are evidence of large-scale vortex dynamics.

Fig. 2 Contours of instantaneous density, ρ/ρ₀, in a spanwise plane for three different lower-to-upper stream density ratios: (a) s = 1.0, (b) s = 1.05, and (c) s = 1.35.

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To validate the mixing layer simulations, flow field results from the s = 1.0 simulation are presented against those from comparable simulations: the previously mentioned weakly compressible DNS by Pantano & Sarkar (case A3) and the DNS of an incompressible mixing layer at an initial momentum thickness Reynolds number of 800 performed by Rogers & Moser [42] (presented in their paper as the baseline simulation). Unlike the current simulation and the simulation by Pantano & Sarkar, Rogers & Moser imposed initial conditions from boundary layer computations to initialize their flow field. The flow field data presented in the following figures have been time-averaged over a period from tΔU/δ_θ(0) = 264.0 to 517.5 when the flow is in the self-similar regime. This is comparable to the timespan over which Pantano & Sarkar averaged their flow-field results, tΔU/δ_θ(0) from 261 to 518. In addition to time-averaging, flow-field results were spanwise, streamwise, and ensemble averaged across the three simulations.

Figure 3 shows the mean streamwise velocity against the transverse coordinate, z, normalized by the vorticity thickness of the shear layer, δ_ω = ΔU/(∂Ū/∂z)_max. Experimental results from investigations of the self-similar region of incompressible shear layers performed by Bell & Mehta [43] and Spencer & Jones [44] are also presented. The velocity profile compares very well with Pantano & Sarkar’s DNS result and reasonably well with experimental measurements. Figure 4 shows a comparison of the rms streamwise, transverse, and spanwise velocity fluctuations and the square root of Reynolds shear stress, ${[\bar{u^{'} w^{'}}]}^{1 / 2}$ , from the s = 1.0 simulation against previous computational and experimental results. Even though the current simulation is performed at Re_θ,0 ten times the previous DNS value and uses subgrid scale modeling, the rms and Reynolds stress profiles compare well with previous calculations. The peak values also compare well for all quantities except for the peak streamwise velocity rms which is approximately 5–10% larger than those from the previous simulations. In all, the current simulation accurately captures the characteristics of the mixing layer and can be used to analyze the aero-optical properties of the flow. Since the other two simulations for s = 1.05 and 1.35 use the same mesh and computational parameters except for the initial temperature and density profiles, all of the simulations are expected to be able to capture the flow and optical physics faithfully.

Fig. 3 Self-similar mean streamwise velocity profiles: ——, s = 1.0 simulation; oe-27-4-5670-i001 , A3 Pantano & Sarkar [38]; – ○ –, Spencer & Jones [44]; oe-27-4-5670-i002 , Bell & Mehta [43].

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Fig. 4 Root-mean-square values of (a) streamwise, (b) transverse, and (c) spanwise velocity fluctuations, and (d) the square-root of Reynolds shear stress across the s = 1.0 shear layer: ——, s = 1.0 simulation; oe-27-4-5670-i003 , A3 Pantano & Sarkar [38]; oe-27-4-5670-i004 , Rogers & Moser [42]; – ○ –, Spencer & Jones [44]; oe-27-4-5670-i005 , Bell & Mehta [43].

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3.3. Aero-optical results

In Fig. 5(a), the streamwise wavenumber spectra of density fluctuations in the shear layer center-plane are shown for the three s values. The spectra are averaged in the spanwise direction and ensemble-averaged over three simulations in each case. Although this flow is non-stationary in time, to improve statistical convergence in the inertial subrange the spectra are time-averaged over a small window from tΔU/δ_θ(0) = 290.5 to 334.9, approximately one-third of a large-scale eddy turnover time. For clarity, the s = 1.05 data in Fig. 5(a) has been multiplied by 10^0.5. The density matched shear layer, s = 1.0, appears to give m_ρ = 5.5/3. For s = 1.05, the value of m_ρ decreases to 4.75/3 and it drops further to 4/3 in the s = 1.35 case. The latter corresponds to the expected spectral slope for temperature in a shear flow at low Reynolds numbers [29].

Fig. 5 Streamwise wavenumber spectra of (a) density at z/δ_θ(0) = 0 and (b) normalized optical path difference for mixing layers with (i) s = 1.0, (ii) s = 1.05 and (iii) s = 1.35. For clarity, in (a), the s = 1.05 data has been multiplied by 10^0.5; in (b), the s = 1.05 data has been multiplied by 10^0.75 and s = 1.35 data has been multiplied by 10^0.5. Dotted lines in each figure reflect the specified power-law slopes.

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From Eq. (19), the optical phase distortions are expected to exhibit a spectral slope of −m_OPD = −(m_ρ + 1). To calculate the OPL, the index of refraction is calculated from the density field obtained from the LES and is then integrated across the mixing layer (along the z-direction in Fig. 1). Using the entire x–y domain as the beam aperture, the spatial mean of the OPL at each time step is then removed to calculate OPD. Figure 5(b) shows the streamwise wavenumber spectra of the normalized optical path difference, ${OPD}^{N} = OPD / [K_{GD} ρ_{0} M_{c}^{2} δ_{θ} (0)]$ for the three shear layers. The normalization is similar to that of Gordeyev et al. [45]. For clarity, in Fig. 5(b) the s = 1.05 data has been multiplied by 10^0.75 and s = 1.35 data has been multiplied by 10^0.5. These OPD spectra confirm that for each case the spectral roll-off slope is indeed equal to −(m_ρ + 1) (equivalently $- (m_{ρ}^{3 D} - 1)$ for homogeneous flow) in the same inertial subrange of wavenumbers as in the density spectra.

These results indicate that depending on the relative strengths of turbulent temperature and pressure fluctuations, the spectral behavior of aero-optical distortions can vary and the use of theories derived for optical distortions by the atmosphere would be incorrect when pressure fluctuations are appreciable. They also imply that even though the flow-field is not homogeneous as was assumed in the theoretical development for the aero-optical phase-distortion spectrum, the relationship between the inertial subrange behavior of turbulent density fluctuations and OPD still holds at high wavenumbers given that turbulence in the inertial subrange is more homogenous and isotropic and thus does not vary strongly along the propagation path.

4. Conclusions

Based on the Gladstone-Dale relation and the ideal gas law, general expressions for the wavenumber spectra of index-of-refraction and aero-optical phase distortions are derived. The results indicate that if the three-dimensional spectrum of turbulent index-of-refraction or density fluctuations has an inertial subrange that follows a power law slope of $- m_{ρ}^{3 D}$ , the one-dimensional phase distortion or OPD spectrum will have a slope of $- (m_{ρ}^{3 D} - 1)$ . Equivalently, if the one-dimensional density or index-of-refraction spectrum has a power law slope of −m_ρ in the inertial subrange, the one-dimensional phase distortion spectrum will have a slope of −(m_ρ + 1). For high-Reynolds-number, weakly-compressible turbulent flows, the spectral behavior of phase distortions is expected to depend on a combination of effects from temperature, pressure, and pressure-temperature cospectra. If the turbulence is homogeneous, temperature fluctuations contribute to the three-dimensional spectrum in the inertial subrange with a power law spectral slope of −11/3, pressure fluctuations contribute a power law spectral slope of −13/3, and pressure-temperature cospectra will have an unknown deleterious contribution. This means that $- m_{ρ}^{3 D}$ can have a spectral slope of −11/3, −13/3, or a value in-between.

To validate theoretical results, large-eddy simulations of three weakly-compressible mixing layers are performed: one where pressure fluctuations dominate the optical distortions, another where temperature fluctuations dominate, and a third at a condition close to realistic separated shear layers observed in recent computational studies. This is accomplished by having the ratio of the bottom stream to top stream density equal to 1.0, 1.35, and 1.05, respectively. It is shown that the one-dimensional spectral slope of OPD for all three mixing layers obeys the theoretical prediction, m_OPD = m_ρ + 1. The value of m_ρ is affected by the flow conditions; the spectral slopes of the mismatched density shear layers are noticeably smaller than that of the matched density shear layer.

The current work presents a physical interpretation for the log-slope behavior seen in experimental measurements and computational predictions of aero-optical distortion spectra. The application of these results can aid in the development of phenomenological understanding of aero-optical distortions. For example, studies into the use of wall cooling to mitigate aero-optical distortions in turbulent boundary layers can potentially use the spectral log-slope of OPD to gauge the relative influence of temperature and pressure on density fluctuations. At a modeling level, the current work provides a more accurate representation of the OPD log-slope for homogeneous aero-optical flows and a key component for modeling the temporal and spatial effects of aero-optical distortions on beam performance.

Funding

National Science Foundation (OCI-0725070 and ACI-1238993); High Energy Laser Joint Technology Office (HEL-JTO) through Air Force Office of Scientific Research grant FA9550-13-1-0001.

Acknowledgments

This work was made possible through the Blue Waters Graduate Fellowship and is a part of the Blue Waters sustained-petascale computing project, which is supported by the National Science Foundation (awards OCI-0725070 and ACI-1238993) and the State of Illinois. Blue Waters is a joint effort of the University of Illinois at Urbana-Champaign and its National Center for Supercomputing Applications. Part of this work in an earlier version was presented in AIAA Paper 2017-3834, 48th AIAA Plasmadynamics and Lasers Conference, Denver, Colorado, 5–9 June 2017.

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