Turbulent energy flux generated by shock/homogeneous-turbulence interaction

2016 ◽  
Vol 796 ◽  
pp. 113-157 ◽  
Author(s):  
Russell Quadros ◽  
Krishnendu Sinha ◽  
Johan Larsson
Keyword(s):  
Shock Wave ◽  
Turbulent Flows ◽  
Energy Flux ◽  
High Speed ◽  
Isotropic Turbulence ◽  
Interaction Analysis ◽  
Turbulent Energy ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Linear Interaction

High-speed turbulent flows with shock waves are characterized by high localized surface heat transfer rates. Computational predictions are often inaccurate due to the limitations in modelling of the unclosed turbulent energy flux in the highly non-equilibrium regions of shock interaction. In this paper, we investigate the turbulent energy flux generated when homogeneous isotropic turbulence passes through a nominally normal shock wave. We use linear interaction analysis where the incoming turbulence is idealized as being composed of a collection of two-dimensional planar vorticity waves, and the shock wave is taken to be a discontinuity. The nature of the postshock turbulent energy flux is predicted to be strongly dependent on the angle of incidence of the incoming waves. The energy flux correlation is also decomposed into its vortical, entropy and acoustic contributions to understand its rapid non-monotonic variation behind the shock. Three-dimensional statistics, calculated by integrating two-dimensional results over a prescribed upstream energy spectrum, are compared with available data from direct numerical simulations. A detailed budget of the governing equation is also considered in order to gain insight into the underlying physics.

Evolution of enstrophy in shock/homogeneous turbulence interaction

2012 ◽  
Vol 707 ◽  
pp. 74-110 ◽  
Author(s):  
Krishnendu Sinha
Keyword(s):  
Shock Wave ◽  
Numerical Simulations ◽  
High Speed ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Interaction Analysis ◽  
Direct Numerical Simulations ◽  
Homogeneous Isotropic Turbulence ◽  
Mean Flow ◽  
Linear Interaction

AbstractInteraction of turbulent fluctuations with a shock wave plays an important role in many high-speed flow applications. This paper studies the amplification of enstrophy, defined as mean-square fluctuating vorticity, in homogeneous isotropic turbulence passing through a normal shock. Linearized Navier–Stokes equations written in a frame of reference attached to the unsteady shock wave are used to derive transport equations for the vorticity components. These are combined to obtain an equation that describes the evolution of enstrophy across a time-averaged shock wave. A budget of the enstrophy equation computed using results from linear interaction analysis and data from direct numerical simulations identifies the dominant physical mechanisms in the flow. Production due to mean flow compression and baroclinic torques are found to be the major contributors to the enstrophy amplification. Closure approximations are proposed for the unclosed correlations in the production and baroclinic source terms. The resulting model equation is integrated to obtain the enstrophy jump across a shock for a range of upstream Mach numbers. The model predictions are compared with linear theory results for varying levels of vortical and entropic fluctuations in the upstream flow. The enstrophy model is then cast in the form of$k$–$\epsilon $equations and used to compute the interaction of homogeneous isotropic turbulence with normal shocks. The results are compared with available data from direct numerical simulations. The equations are further used to propose a model for the amplification of turbulent viscosity across a shock, which is then applied to a canonical shock–boundary layer interaction. It is shown that the current model is a significant improvement over existing models, both for homogeneous isotropic turbulence and in the case of complex high-speed flows with shock waves.

Turbulence structure behind the shock in canonical shock–vortical turbulence interaction

2014 ◽  
Vol 756 ◽  
Author(s):  
Jaiyoung Ryu ◽  
Daniel Livescu
Keyword(s):  
Shock Wave ◽  
Mach Number ◽  
Isotropic Turbulence ◽  
Interaction Analysis ◽  
Longitudinal Velocity ◽  
Wave Structure ◽  
Turbulence Structure ◽  
Turbulent Structures ◽  
Linear Interaction ◽  
Third Invariant

AbstractThe interaction between vortical isotropic turbulence (IT) and a normal shock wave is studied using direct numerical simulation (DNS) and linear interaction analysis (LIA). In previous studies, agreement between the simulation results and the LIA predictions has been limited and, thus, the significance of LIA has been underestimated. In this paper, we present high-resolution simulations which accurately solve all flow scales (including the shock-wave structure) and extensively cover the parameter space (the shock Mach number, $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}M_s$, ranges from 1.1 to 2.2 and the Taylor Reynolds number, ${\mathit{Re}}_{\lambda }$, ranges from 10 to 45). The results show, for the first time, that the turbulence quantities from DNS converge to the LIA solutions as the turbulent Mach number, $M_t$, becomes small, even at low upstream Reynolds numbers. The classical LIA formulae are extended to compute the complete post-shock flow fields using an IT database. The solutions, consistent with the DNS results, show that the shock wave significantly changes the topology of the turbulent structures, with a symmetrization of the third invariant of the velocity gradient tensor and ($M_s$-mediated) of the probability density function (PDF) of the longitudinal velocity derivatives, and an $M_s$-dependent increase in the correlation between strain and rotation.

Measurement on the Structure of the Reynolds Stress for the Turbulent Boundary Layer Flow Over a Two-Dimensional Escarpment With Mild Upwind Slope

2011 ◽  
Author(s):  
Bao-Shi Shiau ◽  
Ben-Jue Tsai
Keyword(s):  
Boundary Layer ◽  
Turbulent Boundary Layer ◽  
Reynolds Stress ◽  
High Speed ◽  
Slope Angle ◽  
Turbulent Energy ◽  
Quadrant Analysis ◽  
Two Dimensional ◽  
Spectrum Distribution ◽  
Layer Flow

Experimental measurement study on the structure of the Reynolds stress and turbulence spectrum for wind flows over a two-dimensional escarpment with mild upwind slope (slope angle θ = 15°) were performed in the wind tunnel. The Quadrant analysis was applied to analyze the experimental data and yield the structure of the Reynolds stress. In according to the quadrant analysis, the Reynolds stress is composed of four events of the stress components, i.e. outward interaction, ejection (low-speed fluid upward), inward interaction, and sweep (high-speed fluid downward). Measured results show that: (1) Measurements of the structure of the Reynolds stress reveal that both the sweep and ejection events are the major contributors to the Reynolds stress for flow around the two dimensional escarpment with mild upwind slope. (2) The contributions to the Reynolds stress made by ejection events and sweep events are almost the same at heights Z/Zref greater than 0.2 for different downstream distances along the mild slope of escarpment. Here Zref is the turbulent boundary layer thickness. When flow reached the top of the slope of escarpment, stress fractions of ejection event and sweep event, S2 and S4 increased significantly. (3) The he turbulent energy spectrum distribution was not found very dominant spectrum peak as winds flow over the mild upwind slope and top surface of escarpment.

scholarly journals The effect of compressibility on the lift of an aerofoil

1928 ◽  
Vol 118 (779) ◽  
pp. 113-119 ◽  
Keyword(s):  
Drag Coefficient ◽  
High Speed ◽  
Lift Coefficient ◽  
Experimental Investigations ◽  
Angle Of Incidence ◽  
Two Dimensional ◽  
Velocity Of Sound ◽  
Similar Calculation ◽  
Incompressible Medium ◽  
Observed Behaviour

At ordinary aeroplane speeds the effect of the compressibility of the air is very small, and there is complete justification for the usual assumption that the air may be regarded effectively as an incompressible medium. This assumption, however, ceases to be valid in the case of high tip-speed airscrews and is not really satisfactory even when the tip speed is no greater than 800 f. p. s. It is important, therefore, to examine, both theoretically and experimentally, the effect of compressibility at high speed on the characteristics of an aerofoil. Experimental investigations are in progress at the Royal Aircraft Establishment in which the aerofoil characteristics are derived by analysing the observed behaviour of high-speed model airscrews, but owing to the complexity both of the experiments and of the analysis it is impossible that the results should have the same accuracy as those obtained from direct tests of an aerofoil at low speed. An attempt has now been made to estimate theoretically the effect of compressibility on the lift of an aerofoil in two-dimensional motion and to indicate the nature of the variation which may be anticipated in the curve of lift coefficient against angle of incidence. It is unfortunately impossible at the present state of knowledge to make any similar calculation for the drag of the aerofoil, but on general grounds we may anticipate that the drag coefficient will rise at an increasing rate until the velocity of sound is reached, and that above this speed the drag coefficient will decrease again, remaining, however, higher than at low speeds.

Helical turbulence and absolute equilibrium

1973 ◽  
Vol 59 (4) ◽  
pp. 745-752 ◽  
Author(s):  
Robert H. Kraichnan
Keyword(s):  
Energy Transfer ◽  
Isotropic Turbulence ◽  
Negative Temperature ◽  
Inviscid Flow ◽  
Turbulent Energy ◽  
Navier Stokes ◽  
Two Dimensional ◽  
Navier Stokes Equation ◽  
Helical Turbulence ◽  
The Absolute

The interaction of two pure helical (circularly polarized) velocity waves according to the incompressible Navier–Stokes equation produces modulation products of mixed helicity. In general, the interaction of waves of opposite helicity is stronger than that of waves with the same helicity. The inference is that strong net helicity depresses overall turbulent energy transfer. The conservation laws strongly inhibit energy transfer from higher to lower wavenumbers, when the helicity is large. The absolute equilibrium spectra of velocity and helicity for an inviscid flow system truncated at an upper wavenumber k2 are \[ U(k) = 2\alpha/(\alpha^2-\beta^2k^2),\quad Q(k) = 2\beta k^2/(\alpha^2-\beta^2k^2), \] where the velocity variance and helicity/unit volume are ∫U(k)d3k and ∫Q(k)d3k, respectively. The temperature parameters α and β are constrained by α > 0 and |βk2| < α. There are no analogues of the negative-temperature equilibrium states known for two-dimensional inviscid flow. It is argued that the inertial-range energy cascade in isotropic turbulence driven by helical input should not differ asymptotically from that of non-helical turbulence. The absolute equilibrium distributions suggest that, in contrast to the analogous two-dimensional situation, statistically steady helical input at middle wavenumbers should not produce a significant downward cascade of energy to lower wavenumbers.

The wall region in turbulent shear flow

1972 ◽  
Vol 54 (1) ◽  
pp. 39-48 ◽  
Author(s):  
James M. Wallace ◽  
Helmut Eckelmann ◽  
Robert S. Brodkey
Keyword(s):  
Turbulent Flows ◽  
Reynolds Stress ◽  
Time Scales ◽  
High Speed ◽  
Turbulent Energy ◽  
Turbulent Shear ◽  
Wall Region ◽  
Turbulent Shear Flow ◽  
Sequence Of Events ◽  
Hot Film

Hot-film measurements in a fully developed channel flow have been made in an attempt to gain more insight into the process of Reynolds stress production. The background for this effort is the observation of a certain sequence of events (deceleration, ejection and sweep) in the wall region of turbulent flows by Corino (1965) and Corino & Brodkey (1969). The instantaneous product signal uv was classified according to the sign of its components u and v, and these classified portions were then averaged to obtain their contributions to the Reynolds stress $-\rho\overline{uv} $. The signal was classified into four categories; the two main ones were that with u negative and v positive, which can be associated with the ejection-type motion of Corino & Brodkey (1969), and that with u positive and v negative, associated with the sweep-type motion. It was found that over the wall region investigated, 3·5 [les ] y [les ] 100, these two types of motion give rise to a stress considerably greater than the total Reynolds stress. Two other types of motion, (i) u negative, v negative, corresponding to low-speed fluid deflected towards the wall, and (ii) u positive, v positive, corresponding to high-speed fluid reflected outwards from the wall, were found to account for the ‘excess’ stress produced by the first two categories, which give contributions of opposite sign.The autocorrelations of the classified portions of uv were obtained to determine the relative time scales of these four types of motion. The positive stress producing motions (u < 0, v > 0 and u > 0, v < 0) were found to have significantly larger time scales than the negative stress producing motions (u < 0, v < 0 and u > 0, v > 0). It was further surmised that turbulent energy dissipation is associated with the Reynolds stress producing motions, since these result in localized shear regions in which the dissipation is several orders of magnitude greater than the average dissipation at the wall.

scholarly journals Analytical Solution of Heat Conduction from Turbulence with an Isotropic Example

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Steven A. E. Miller
Keyword(s):  
Power Spectrum ◽  
Heat Equation ◽  
Turbulent Flows ◽  
High Speed ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Single Equation ◽  
Aerodynamic Heating ◽  
Acoustic Analogy ◽  
Chaotic Motions

Aerodynamic heating due to turbulence significantly affects the operation of high-speed vehicles and the entrainment of fluid by turbulent plumes. In this paper, the heat generated and convected by fluid turbulence is examined by rearranging the Navier-Stokes equations into a single equation for the fluctuating dependent variables external to unsteady chaotic motions. This equation is similar to the nonhomogeneous heat equation where sources are terms resulting from this rearrangement. Mean and fluctuating quantities are introduced, and under certain circumstances, a heat equation for the fluctuating density results with corresponding mean and fluctuating source terms. The resultant equation is similar to Lighthill’s acoustic analogy and is a “heat analogy.” A solution is obtained with the use of Green’s function as long as the observer is located outside the region of chaotic motion. Predictions for the power spectrum are shown for high Reynolds number isotropic turbulence. The power spectrum decays as the inverse of the wavenumber of the turbulent velocity fluctuations. The developed theory can easily be applied to other turbulent flows if the statistics of unsteady motion can be estimated.

Lagrangian Statistics of Droplet Velocity and Acceleration in Isotropic Turbulence

2009 ◽  
Author(s):  
Balaji Gopalan ◽  
Edwin Malkiel ◽  
Joseph Katz
Keyword(s):  
Turbulent Flows ◽  
High Speed ◽  
Isotropic Turbulence ◽  
Droplet Diameter ◽  
Relative Size ◽  
Stokes Number ◽  
Droplet Velocity ◽  
Lagrangian Statistics ◽  
Oceanic Turbulence ◽  
Wide Range

The addition of dispersants, water and oil soluble surfactants that lower the interfacial tension of the crude oil, along with oceanic turbulence can breakdown oil spills into droplets. Knowledge of the dispersion rate of these droplets by oceanic turbulence is essential for the development of better models to assess the environmental impact of spills. The objective of this research is to study, experimentally, the dispersion of oil droplets in turbulent flows. The measurements are performed in a specialized laboratory facility that enables generation of carefully controlled, isotropic, homogeneous turbulence at a wide range of fully characterized intensities and length scales. The oil dispersion is visualized using high-speed inline digital holographic cinematography. Holographic data has been analyzed and Lagrangian statistics of droplet velocity, dispersion and acceleration has been calculated. As the relative size of the droplet diameter to the Kolmogorov length scale and its Stokes number increases, the acceleration autocorrelation shifts from dropping to zero faster than the fluid particles to slower.

scholarly journals Interaction of two-dimensional spots with a heat releasing/absorbing shock wave: linear interaction approximation results

2019 ◽  
Vol 871 ◽  
pp. 865-895 ◽  
Author(s):  
G. Farag ◽  
P. Boivin ◽  
P. Sagaut
Keyword(s):  
Shock Wave ◽  
Compressible Flows ◽  
Two Dimensional ◽  
Extended Version ◽  
Unified Framework ◽  
Linear Interaction ◽  
Planar Shock ◽  
Planar Shock Wave ◽  
Gaussian Disturbance ◽  
Interaction Approximation

The canonical interaction between a two-dimensional weak Gaussian disturbance (entropy spot, density spot, weak vortex) with an exothermic/endothermic planar shock wave is studied via the linear interaction approximation. To this end, a unified framework based on an extended Kovásznay decomposition that simultaneously accounts for non-acoustic density disturbances along with a poloidal–toroidal splitting of the vorticity mode and for heat release is proposed. An extended version of Chu’s definition for the energy of disturbances in compressible flows encompassing multi-component mixtures of gases is also proposed. This new definition precludes spurious non-normal phenomena when computing the total energy of extended Kovásznay modes. Detailed results are provided for three cases, along with fully general expressions for mixed solutions that combine incoming vortical, entropy and density disturbances.

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