Pressure-dilatation Correlation Research Articles

Local geometry of a weak normal shock wave interacting with turbulence

The shock surface geometry is investigated with direct numerical simulations of a weak normal shock wave propagating in turbulence. The geometry is quantified with the principal curvatures of the surface. A large part of the surface has an approximately flat saddle shape, while elliptic concave and convex shapes with a large curvature intermittently appear on the shock surface. The pressure–dilatation correlation in the governing equation of pressure is investigated at the shock wave with the decomposition into three terms associated with the velocity gradients in the two directions of the principal curvatures and the normal direction of the shock wave. Fluid expansion in the tangential direction occurs at the shock wave with a convex shape in the direction of the shock propagation, resulting in a smaller pressure jump across the shock wave. For a concave shape, compression in the tangential direction can amplify the pressure jump. Consistently, small and large shock Mach numbers are observed for convex and concave shapes, respectively. The geometric influences are the most significant for elliptic concave and convex shapes with approximately equal curvatures in the two principal directions because the compression or expansion occurs in all tangential directions. These relations between the shock surface geometry and shock Mach number observed in turbulence are consistent with the theory of deformed shock waves, suggesting that the three-dimensional geometrical features of the shock surface are important in the modulation of shock waves due to turbulence.

Assessment of turbulence models using DNS data of compressible plane free shear layer flow

The present paper uses the detailed flow data produced by direct numerical simulation (DNS) of a three-dimensional, spatially developing plane free shear layer to assess several commonly used turbulence models in compressible flows. The free shear layer is generated by two parallel streams separated by a splitter plate, with a naturally developing inflow condition. The DNS is conducted using a high-order discontinuous spectral element method (DSEM) for various convective Mach numbers. The DNS results are employed to provide insights into turbulence modelling. The analyses show that with the knowledge of the Reynolds velocity fluctuations and averages, the considered strong Reynolds analogy models can accurately predict temperature fluctuations and Favre velocity averages, while the extended strong Reynolds analogy models can correctly estimate the Favre velocity fluctuations and the Favre shear stress. The pressure–dilatation correlation and dilatational dissipation models overestimate the corresponding DNS results, especially with high compressibility. The pressure–strain correlation models perform excellently for most pressure–strain correlation components, while the compressibility modification model gives poor predictions. The results of an a priori test for subgrid-scale (SGS) models are also reported. The scale similarity and gradient models, which are non-eddy viscosity models, can accurately reproduce SGS stresses in terms of structure and magnitude. The dynamic Smagorinsky model, an eddy viscosity model but based on the scale similarity concept, shows acceptable correlation coefficients between the DNS and modelled SGS stresses. Finally, the Smagorinsky model, a purely dissipative model, yields low correlation coefficients and unacceptable accumulated errors.

Fluctuations of thermodynamic variables in compressible isotropic turbulence laden with inertial particles

The modifications of thermodynamic fluctuations by inertial particles are investigated in decaying compressible isotropic turbulence with an initial turbulent Mach number of 1.2 through direct numerical simulations. The particles interact with turbulence through two-way coupling under the Eulerian–Lagrangian point-source framework. Five simulations with different particle diameters (Stokes numbers) are conducted and compared with the particle-free simulation. The underlying modulation mechanisms are revealed through analyzing the transport equations of thermodynamic variables. The fluctuation features are similar for density, pressure, and temperature in compressible isotropic turbulence. The addition of particles enhances the mean pressure and temperature, and the enhancement becomes more significant with the increment of the Stokes numbers. Nevertheless, the thermodynamic fluctuations are attenuated, and the attenuation is greater for larger particles. Additionally, the thermodynamic fluctuations deviate from isentropic behavior in compressible turbulence, and the deviations are further augmented by the inertial particles. As the particle inertia increases, the departures from isentropic fluctuations gradually increase at small scales, but decrease at large scales. The decay of thermodynamic fluctuations is dominated by the correlation between thermodynamic variables and dilatation. Because of the high inertia, the particles retain their kinetic energy longer than the surrounding fluid and continuously perform positive work on the internal energy of fluid. The mean pressure and pressure-dilatation correlation are thus augmented, resulting in the attenuation of the fluctuations of pressure and other thermodynamic variables.

On the near-wall structures and statistics of fluctuating pressure in compressible turbulent channel flows

The near-wall structures and statistics of fluctuating pressure (p′) in compressible turbulent channel flows (CTCF) with isothermal walls have been investigated by direct numerical simulations. Two typical cases for high bulk Mach number Ma = 3.83 and low one Ma = 1.56 are considered. A novel type of near-wall pressure structures named “alternating positive and negative structures (APNS)” is found in the high-Ma case based on the comprehensive analysis of spectra and dynamic mode decomposition of p′. These APNS of p′ are identified to have the streamwise and spanwise length scales of (λx/h, λz/h) ≈ (0.9, 1.5), where h is the channel half-height, and prefer to inhabit the low-speed wall streaks. It is also verified via a pressure splitting method that the APNS of p′ are dominated by the compressibility effects. Based on the linear stability analysis, the APNS of p′ can be intimately related to a linear stability eigenmode of the high-Ma CTCF and are sustained by the transient growth mechanism as the disturbances of the APNS length scales. Furthermore, these APNS of p′ offer an extra mechanism to generate the near-wall p′ for the high-Ma case. Moreover, it is found that the APNS of p′ have a dominating effect on the pressure-dilatation correlation and the production of Reynolds shear stress. The present study may provide a reliable way to achieve a better understanding and modeling of compressibility effects in the wall-bounded turbulence of high Ma.

Heat release effects on the Reynolds stress budgets in turbulent premixed jet flames at low and high Karlovitz numbers

In turbulent premixed flames at low Karlovitz number, combustion heat release can have a significant impact on turbulence. Thermal expansion in flame induces dilatation, and the corresponding pressure–dilatation correlation acts as a primary source of turbulent kinetic energy (TKE). As a consequence, the flame-normal component of the normal Reynolds stresses significantly increases. Additionally, for sheared flames, typical of jet flames, the shear component of the Reynolds stresses exhibits counter-Boussinesq behavior. For flames at low Karlovitz number, where these effects dominate, no models have successfully predicted all Reynolds stress components. To develop more complete turbulence models, heat release effects on the evolution of all Reynolds stress components need to be analyzed. In this work, Reynolds stress budgets are evaluated from Direct Numerical Simulation (DNS) databases of spatially-evolving turbulent premixed planar jet flames at low and high Karlovitz numbers. In the Reynolds stress budgets, the velocity–pressure gradient correlation term is important at both Karlovitz numbers but serves fundamentally different roles in each case. In the budgets for the normal components, the velocity–pressure gradient correlation term is decomposed into a redistribution term and an isotropic term, where the redistribution term acts to redistribute energy between the Reynolds stress components and the isotropic term is the pressure–dilatation source term of turbulent kinetic energy. At high Karlovitz number, the isotropic term is negligible, and the redistribution term acts to isotropize the turbulence as in non-reacting flows. Conversely, at low Karlovitz number, the isotropic term acts as a large source, and the redistribution term preferentially injects energy into the flame-normal component at the expense of other components which acts to make the turbulence less isotropic. In the budget for the shear component, the shear production term dominates the velocity–pressure gradient correlation term at high Karlovitz number, but the opposite is observed at low Karlovitz number. The dominance of the velocity–pressure gradient correlation term at low Karlovitz number is primarily induced by the flame-generated mean pressure gradient and ultimately leads to the counter-Boussinesq behavior of the shear component. The overall analysis indicates that any turbulence model that relies on small-scale isotropy and/or rapid isotropization will fail to capture the heat release effects on turbulence at low Karlovitz number.

Large-eddy simulation of shock-turbulence interaction in supersonic diffuser flows

ABSTRACTLES based on explicit filtering is used to study the shock train phenomenon in turbulent supersonic diffuser flows with circular cross-section and isothermal wall with an incoming pipe flow at friction Reynolds number 245 and centerline Mach number 1.7. Alternate regions of compression and expansion are found in the shock train which is followed by a shock-free ‘mixing’ region as observed in experiments and simulations in the literature. Turbulence amplification and local peaks in pressure-dilatation correlation are observed in the vicinity of the shocks. Low-frequency oscillations of the shock train are also observed.

Reynolds and Mach number scaling in solenoidally-forced compressible turbulence using high-resolution direct numerical simulations

We report results from direct numerical simulation (DNS) of stationary compressible isotropic turbulence at very-high resolutions and a range of parameters using a massively parallel code at Taylor Reynolds numbers ($R_{{\it\lambda}}$) ranging from $R_{{\it\lambda}}=38$ to $430$ and turbulent Mach numbers ($M_{t}$) ranging from 0.1 to 0.6 on up to $2048^{3}$ grid resolutions. A stationary state is maintained by a stochastic solenoidal forcing at the largest scales. The focus is on the mechanisms of energy exchanges, namely, dissipation, pressure-dilatation correlation and the individual contributing variables. Compressibility effects are studied by decomposing velocity and pressure fields into solenoidal and dilatational components. We suggest a critical turbulent Mach number at about 0.3 that separate two different flow regimes – only at Mach numbers above this critical value do we observe dilatational effects to affect the flow behaviour in a qualitative manner. The equipartition of energy between the dilatational components of kinetic and potential energy, originally proposed for decaying flows at low $M_{t}$, presents significant scatter at low $M_{t}$, but appears to be valid at high $M_{t}$ for stationary flows, which is explained by the different role of dilatational pressure in decaying and stationary flows, and at low and high $M_{t}$. While at low $M_{t}$ pressure possesses characteristics of solenoidal pressure, at high $M_{t}$ it behaves in similar ways to dilatational pressure, which results in significant changes in the dynamics of energy exchanges. This also helps explain the observed qualitative change in the skewness of pressure at high $M_{t}$ reported in the literature. Regions of high pressure are found to be correlated with regions of intense local expansions. In these regions, the density–temperature correlation is also seen to be relatively high. Classical scaling laws for low-order moments originally proposed for incompressible turbulence appear to be only weakly affected by compressibility for the range of $R_{{\it\lambda}}$ and $M_{t}$ investigated.

On velocity and reactive scalar spectra in turbulent premixed flames

AbstractKinetic energy and reactive scalar spectra in turbulent premixed flames are studied from compressible three-dimensional direct numerical simulations (DNS) of a temporally evolving rectangular slot-jet premixed flame, a statistically one-dimensional configuration. The flames correspond to a lean premixed hydrogen–air mixture at an equivalence ratio of 0.7, preheated to 700 K and at 1 atm, and three DNS are considered with a fixed jet Reynolds number of 10 000 and a jet Damköhler number varying between 0.13 and 0.54. For the study of spectra, motivated by the need to account for density change, which can be locally strong in premixed flames, a new density-weighted definition for two-point velocity/scalar correlations is proposed. The density-weighted two-point correlation tensor retains the essential properties of its constant-density (incompressible) counterpart and recovers the density-weighted Reynolds stress tensor in the limit of zero separation. The density weighting also allows the derivation of balance equations for velocity and scalar spectrum functions in the wavenumber space that illuminate physics unique to combusting flows. Pressure–dilatation correlation is a source of kinetic energy at high wavenumbers and, analogously, reaction rate–scalar fluctuation correlation is a high-wavenumber source of scalar energy. These results are verified by the spectra constructed from the DNS data. The kinetic energy spectra show a distinct inertial range with a $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}-5/3$ scaling followed by a ‘diffusive–reactive’ range at higher wavenumbers. The exponential drop-off in this range shows a distinct inflection in the vicinity of the wavenumber corresponding to a laminar flame thickness, $\delta _L$, and this is attributed to the contribution from the pressure–dilatation term in the energy balance in wavenumber space. Likewise, a clear spike in spectra of major reactant species (hydrogen) arising from the reaction-rate term is observed at wavenumbers close to $\delta _L$. It appears that in the inertial range classical scaling laws for the spectra involving the Kolmogorov scale are applicable, but in the high-wavenumber range where chemical reactions have a strong signature the laminar flame thickness produces a better collapse. It is suggested that a full scaling should perhaps involve the Kolmogorov scale, laminar flame thickness, Damköhler number and Karlovitz number.

An Improved K-ω Model for High Speed Flows

An improved K-ω model, which allows for compressible corrections, is proposed in this paper. Numerical scheme was established utilizing the improved Total Variation Diminishing (TVD) scheme and applying implicit scheme to the negative source terms of the turbulence model. Hypersonic flat-plate boundary-layer flows and hypersonic compression ramp flows marked with separation, reattachment and shock/boundary layer interactions are then computed. Comparisons between the computational results, the experimental results and the semi-empirical formulations show that the compressible correction term of the K-ω turbulence model is a pressure-dilatation correlation. In addition, for flow with separation and without separation, calculation results of wall pressures, friction coefficients and wall heat transfer rate distributions using the improved model and established scheme agree better with the experimental results than that using the original K-ω model.

Numerical investigation of the compressible flow past an aerofoil

Numerical investigation of the compressible flow past an 18% thick circular-arc aerofoil was carried out using detached-eddy simulation for a free-stream Mach numberM∞= 0.76 and a Reynolds numberRe= 1.1 × 107. Results have been validated carefully against experimental data. Various fundamental mechanisms dictating the intricate flow phenomena, including moving shock wave behaviours, turbulent boundary layer characteristics, kinematics of coherent structures and dynamical processes in flow evolution, have been studied systematically. A feedback model is developed to predict the self-sustained shock wave motions repeated alternately along the upper and lower surfaces of the aerofoil, which is a key issue associated with the complex flow phenomena. Based on the moving shock wave characteristics, three typical flow regimes are classified as attached boundary layer, moving shock wave/turbulent boundary layer interaction and intermittent boundary layer separation. The turbulent statistical quantities have been analysed in detail, and different behaviours are found in the three flow regimes. Some quantities, e.g. pressure-dilatation correlation and dilatational dissipation, have exhibited that the compressibility effect is enhanced because of the shock wave/boundary layer interaction. Further, the kinematics of coherent vortical structures and the dynamical processes in flow evolution are analysed. The speed of downstream-propagating pressure waves in the separated boundary layer is consistent with the convection speed of the coherent vortical structures. The multi-layer structures of the separated shear layer and the moving shock wave are reasonably captured using the instantaneous Lamb vector divergence and curl, and the underlying dynamical processes are clarified. In addition, the proper orthogonal decomposition analysis of the fluctuating pressure field illustrates that the dominated modes are associated with the moving shock waves and the separated shear layers in the trailing-edge region. The results obtained in this study provide physical insight into the understanding of the mechanisms relevant to this complex flow.

A study of compressibility effects on structure of homogeneous sheared turbulence

A study of compressibility effects on structure and evolution of a sheared homogeneous turbulent flow is carried out using rapid distortion theory (RDT). The focus of this paper consists in validating RDT code by considering the direct numerical simulation (DNS) and RDT results of Simone [Simone, Coleman and Cambon, Fluid Mech. 330 , 307 (1997)]. We will use this theory to clarify the physics of the compressible turbulent flows. An analysis of the behavior of different terms appearing in the turbulent kinetic energy and the Reynolds stress equations permit to well identify compressibility effects which allow us to analyze performance of the compressible model of Fujiwara and Arakawa concerning the pressure-dilatation correlation. The evaluation of this model stays in the field of RDT validity.

Effects of pressure fluctuations on turbulence growth in compressible homogeneous shear flow

Direct numerical simulation results are used to study compressibility effects on turbulent energy growth in homogeneous shear flow. Normalized amplitude of the pressure fluctuations is shown to decrease as the turbulent Mach number increases. This decrease causes the reduction in the pressure–strain term, changes the anisotropy of the Reynolds stress, and reduces the growth rate of turbulent kinetic energy as is the case of mixing layers. The transport equation for the pressure variance is examined in detail. It is shown that when the turbulent Mach number is high the pressure-variance dissipation term is not negligible and contributes to the reduction in the pressure fluctuations. The pressure-variance dissipation is closely related to the high-wavenumber part of the pressure-variance spectrum. Profile of the spectrum is observed to depend on the turbulent Mach number. A statistical theory called the two-scale direct-interaction approximation is applied to obtain model expressions for pressure-related terms. The pressure–dilatation correlation is a dominant term in the pressure-variance equation; its model expression as derived contains the material derivative of turbulent kinetic energy. This quantity represents nonequilibrium properties of turbulent field and can explain the very small value of the pressure dilatation in boundary layers and channel flows. The model expression for the pressure–strain correlation is also derived; the normalized pressure variance can be a parameter that describes compressibility effect on the pressure strain.

Effects of Heat of Reaction on Homogeneous Compressible Turbulence

Direct numerical simulations (DNS) of homogeneous compressible turbulent reacting flows are conducted to investigate the effects of heat of reaction on the solenoidal and the dilatational turbulent motions. Consistent with the previous theoretical results, it is shown that the heat of reaction does not have a significant effect on the low order moments of the solenoidal velocity. However, the variances of dilatation, pressure, temperature, and density, increase significantly due to heat release. Also, the magnitudes of the skewness and the kurtosis of dilatation increase significantly as a result of heat of reaction. The high negative values of the dilatation correspond to localized structures with low probability of existence. The heat of reaction also enhances the amplitude and the frequency of the oscillations of the pressure-dilatation correlation. Examination of the energy transfer among rotational and compressive components of the kinetic energy and the internal energy indicates that the energy of the reaction is transferred to the compressive component of the kinetic energy by the pressure-dilatation correlation. The advection term then transfer the energy from the compressive component of the kinetic energy to its rotational component. While the interactions between compressive component of the kinetic energy and the internal energy are significant, those between the rotational and the compressive components of the kinetic energy are relatively weak.

Towards modeling inhomogeneous compressible turbulence using a two-scale statistical theory

Turbulence models for compressible flows are investigated using a statistical theory called the two-scale direct-interaction approximation. Inertial-range spectra for velocity and density variances are assumed to derive models for several correlations systematically; they include the dilatation dissipation, mass flux, Reynolds stress, and pressure–dilatation correlation. Model expressions are shown to contain two important parameters: the turbulent Mach number and the density variance normalized by the mean density. Typical terms are material derivatives of the turbulent kinetic energy and its dissipation rate as well as the mean velocity divergence. The statistical theory is also applied to a two-time velocity correlation to derive a transport equation for the eddy viscosity. The equation is combined with the turbulent kinetic energy equation to derive a model equation for the dissipation rate. Direct numerical simulation data of isotropic and homogeneous shear turbulence are used to examine models for the dilatation dissipation and the pressure dilatation. The normalized density variance is shown to be useful to explain results from two runs of isotropic turbulence with different initial conditions.

Study of compressibility modifications to the k−ε turbulence model

In modeling the effect of compressibility on the turbulence structure, additional physical mechanisms arise in production and dissipation of turbulent kinetic energy. Several proposed treatments dealing with the dilatation dissipation and the pressure dilatation correlation are discussed in the context of the k−ε two-equation model. Modifications accounting for the turbulent mass flux, enthalpic production, and baroclinic torque are also assessed along with the nonequilibrium treatment. These compressibility models are evaluated with the aid of experimental data for supersonic flow over an axisymmetric afterbody. With the free-stream Mach number of 2.46, the compressibility and nonequilibrium modifications give marginally better results for the turbulence structure and offer added insight into the mean and fluctuating flow fields.

The effect of compressibility on turbulent shear flow: a rapid-distortion-theory and direct-numerical-simulation study

The influence of compressibility upon the structure of homogeneous sheared turbulence is investigated. For the case in which the rate of shear is much larger than the rate of nonlinear interactions of the turbulence, the modification caused by compressibility to the amplification of turbulent kinetic energy by the mean shear is found to be primarily reflected in pressure–strain correlations and related to the anisotropy of the Reynolds stress tensor, rather than in explicit dilatational terms such as the pressure–dilatation correlation or the dilatational dissipation. The central role of a ‘distortion Mach number’ Md = S[lscr ]/a, where S is the mean strain or shear rate, [lscr ] a lengthscale of energetic structures, and a the sonic speed, is demonstrated. This parameter has appeared in previous rapid-distortion-theory (RDT) and direct-numerical-simulation (DNS) studies; in order to generalize the previous analyses, the quasi-isentropic compressible RDT equations are numerically solved for homogeneous turbulence subjected to spherical (isotropic) compression, one-dimensional (axial) compression and pure shear. For pure-shear flow at finite Mach number, the RDT results display qualitatively different behaviour at large and small non-dimensional times St: when St < 4 the kinetic energy growth rate increases as the distortion Mach number increases; for St > 4 the inverse occurs, which is consistent with the frequently observed tendency for compressibility to stabilize a turbulent shear flow. This ‘crossover’ behaviour, which is not present when the mean distortion is irrotational, is due to the kinematic distortion and the mean-shear-induced linear coupling of the dilatational and solenoidal fields. The relevance of the RDT is illustrated by comparison to the recent DNS results of Sarkar (1995), as well as new DNS data, both of which were obtained by solving the fully nonlinear compressible Navier–Stokes equations. The linear quasi-isentropic RDT and nonlinear non-isentropic DNS solutions are in good general agreement over a wide range of parameters; this agreement gives new insight into the stabilizing and destabilizing effects of compressibility, and reveals the extent to which linear processes are responsible for modifying the structure of compressible turbulence.

Modeling Compressibility Effects on Turbulent Flows.

This paper proposes a three-equation (k-ε-F) model for compressible turbulence. Recent direct simulations of homogeneous turbulence show that compressibility effects depend more on the initial conditions than on the turbulent Mach number. This implies that the level of compressibility in turbulence may not be predicted only by solving the equations of k and ε. In this study therefore the variable F, which is defined by the dilatational components of turbulence, is introduced and the equation of F is also solved. We also focus on modeling the pressure-dilatation correlation. An analysis of shocklike structures reveals that this correlation is strongly related to the dilatation dissipation. The proposed model is validated by previous direct simulations of compressible turbulence.

Compressibility effects on the growth and structure of homogeneous turbulent shear flow

Compressibility effects within decaying isotropic turbulence and homogeneous turbulent shear flow have been studied using direct numerical simulation. The objective of this work is to increase our understanding of compressible turbulence and to aid the development of turbulence models for compressible flows. The numerical simulations of compressible isotropic turbulence show that compressibility effects are highly dependent on the initial conditions. The shear flow simulations, on the other hand, show that measures of compressibility evolve to become independent of their initial values and are parameterized by the root mean square Mach number. The growth rate of the turbulence in compressible homogeneous shear flow is reduced compared to that in the incompressible case. The reduced growth rate is the result of an increase in the dissipation rate and energy transfer to internal energy by the pressure–dilatation correlation. Examination of the structure of compressible homogeneous shear flow reveals the presence of eddy shocklets, which are important for the increased dissipation rate of compressible turbulence.

The pressure–dilatation correlation in compressible flows

Simulations of simple compressible flows have been performed to enable the direct estimation of the pressure–dilatation correlation. The generally accepted belief that this correlation may be important in high-speed flows has been verified by the simulations. The pressure–dilatation correlation is theoretically investigated by considering the equation for fluctuating pressure in an arbitrary compressible flow. This leads to the isolation of a component of the pressure–dilatation that exhibits temporal oscillations on a fast time scale. Direct numerical simulations of homogeneous shear turbulence and isotropic turbulence show that this fast component has a negligible contribution to the evolution of turbulent kinetic energy. Then, an analysis for the case of homogeneous turbulence is performed to obtain a formal solution for the nonoscillatory pressure–dilatation. Simplifications lead to a model that algebraically relates the pressure–dilatation to quantities traditionally obtained in incompressible turbulence closures. The model is validated by direct comparison with the simulations.

Rapid distortion theory for homogeneous compressed turbulence with application to modelling

Compressible rapid distortion theory is used to examine pressure fluctuations and the pressure–dilatation correlation in a field of turbulence subjected to rapid homogeneous compression. It is shown how a one-dimensional compression produces large solenoidal pressure fluctuations. As the dimensionality of the compression increases, the magnitude of these fluctuations decreases — it vanishes for a spherically symmetric compression. By contrast the dilatational, or acoustic, pressure fluctuations depend mainly on the net volumetric compression, and are relatively insensitive to the dimensionality of the compression. These same comments apply to the pressure–dilatation correlation.The pressure–dilatation correlation appears in the compressible turbulent kinetic energy equation and is significant in rapidly evolving flows; Reynolds stress closure models require that it be represented. The continuity equation provides a relation between pressure dilatation and the rate of change of pressure fluctuation variance. This relation is the basis for our RDT analysis. That analysis leads to a proposal for modelling the rapid contribution to pressure dilatation.