The formulation of the RANS equations for supersonic and hypersonic turbulent flows

2020 ◽  
pp. 1-31
Author(s):  
H. Zhang ◽  
T.J. Craft ◽  
H. Iacovides
Keyword(s):  
Kinetic Energy ◽  
Pressure Gradient ◽  
Turbulent Kinetic Energy ◽  
Turbulent Flows ◽  
High Speed ◽  
Mean Flow ◽  
Rans Equations ◽  
Flow Equations ◽  
The Mean ◽  
The Impact

Abstract Accurate prediction of supersonic and hypersonic turbulent flows is essential to the design of high-speed aerospace vehicles. Such flows are mainly predicted using the Reynolds-Averaged Navier–Stokes (RANS) approach in general, and in particular turbulence models using the effective viscosity approximation. Several terms involving the turbulent kinetic energy (k) appear explicitly in the RANS equations through the modelling of the Reynolds stresses in such approach, and similar terms appear in the mean total energy equation. Some of these terms are often ignored in low, or even supersonic, speed simulations with zero-equation models, as well as some one- or two-equation models. The omission of these terms may not be appropriate under hypersonic conditions. Nevertheless, this is a widespread practice, even for very high-speed turbulent flow simulations, because many software packages still make such approximations. To quantify the impact of ignoring these terms in the RANS equations, two linear two-equation models and one non-linear two-equation model are applied to the computation of five supersonic and hypersonic benchmark cases, one 2D zero-pressure gradient hypersonic flat plate case and four shock wave boundary layer interaction (SWBLI) cases. The surface friction coefficients and velocity profiles predicted with different combinations of the turbulent kinetic energy terms present in the transport equations show little sensitivity to the presence of these terms in the zero-pressure gradient case. In the SWBLI cases, however, comparisons show that inclusion of k in the mean flow equations can have a strong effect on the prediction of flow separation. Therefore, it is highly recommended to include all the turbulent kinetic energy terms in the mean flow equations when dealing with simulations of supersonic and hypersonic turbulent flows, especially for flows with SWBLIs. As a further consequence, since k may not be obtained explicitly in zero-equation, or certain one-equation, models, it is debatable whether these models are suitable for simulations of supersonic and hypersonic turbulent flows with SWBLIs.

Reynolds Stress Field and Turbulent Kinetic Energy Budget in a Repeating Compressor Stage

2020 ◽  
Author(s):  
Ahmad Nawab ◽  
Feng Wang ◽  
Luca di Mare ◽  
John J. Adamczyk
Keyword(s):  
Kinetic Energy ◽  
Pressure Gradient ◽  
Turbulent Kinetic Energy ◽  
Turbulence Model ◽  
Turbulence Modelling ◽  
Compressor Stage ◽  
Mean Flow ◽  
Rans Simulation ◽  
Dissipation Rates ◽  
The Mean

Abstract Turbulence modelling in compressor passages continues to be a challenging problem. In order to better understand the shortcomings of turbulence modelling, a LES and a RANS computation were performed of a repeating compressor stage. The computation was carried out near the aerodynamic design point of the compressor stage, in order to minimise the challenge posed to the turbulence model. The use of a repeating stage configuration removes the need to specify the statistics of the incoming turbulent field; the statistics become an output of the simulation and not an input. This is a critical fact that greatly increases the credibility of the current LES compressor simulation over many previous simulations. As the computations are performed at mid-span, radial gradients can safely be assumed to be small, thus removing issues associated with capturing flow features attributed to 3D geometry. The flow field is assumed to be incompressible, which is required in order to achieve a true repeating stage environment. The RANS computation is based on a state-of-the-art turbulence model. At the same flow coefficient, the RANS simulation yielded a total pressure rise very near that of the LES simulation. However, there are nontrivial differences in the flow details. The mean flow and Reynolds shear stress boundary layer profiles are in good agreement in regions of favourable pressure gradient, but significant differences exist in the presence of adverse pressure-gradients. The turbulent kinetic energy profiles however are in poor agreement throughout the flow. The mean flow production rates predicted by the RANS computation are largely similar to those of the LES simulation forward of mid-chord where the pressure gradient is favourable. A notable exception is the leading-edge region where the LES predicts negative production i.e. a net transfer of energy to the time-mean flow, and the region aft of mid-chord where the pressure gradient is adverse. Outside of the viscous sub-layer, the dissipation rates are also predicted correctly by the RANS simulation forward of midchord where the pressure gradient is favourable. Aft of mid-chord however, there are significant differences in the dissipation rates.

Near-wall measurements in a three-dimensional turbulent boundary layer

1997 ◽  
Vol 350 ◽  
pp. 189-208 ◽  
Author(s):  
DEBORA A. COMPTON ◽  
JOHN K. EATON
Keyword(s):  
Boundary Layer ◽  
Shear Stress ◽  
Kinetic Energy ◽  
Turbulent Boundary Layer ◽  
Pressure Gradient ◽  
Turbulent Kinetic Energy ◽  
Three Dimensional ◽  
Mean Flow ◽  
The Mean ◽  
Near Wall

An experiment was performed to measure near-wall velocity and Reynolds stress profiles in a pressure-driven three-dimensional turbulent boundary layer. An initially two-dimensional boundary layer (Reθ≈4000) was exposed to a strong spanwise pressure gradient. At the furthest downstream measurement locations there was also a fairly strong favourable streamwise pressure gradient.Measurements were made using a specially designed near-wall laser-Doppler anemometer (LDA), in addition to conventional methods. The LDA used short focal length optics, a mirror probe suspended in the flow, and side-scatter collection to achieve a measuring volume 35 μm in diameter and approximately 65 μm long.The data presented include mean velocity measurements and Reynolds stresses, all extending well below y+=10, at several profile locations. Terms of the turbulent kinetic energy transport equation are presented at two profile locations. The mean flow is nearly collateral (i.e. W is proportional to U) at the wall. Turbulent kinetic energy is mildly suppressed in the near-wall region and the shear stress components are strongly affected by three-dimensionality. As a result, the ratio of shear stress to turbulent kinetic energy is suppressed throughout most of the boundary layer. The angles of stress and strain are misaligned, except very near the wall (around y+=10) where the angles nearly coincide with the mean flow angle. Three-dimensionality appears to mildly reduce the production of turbulent kinetic energy.

Grid Requirements for Multiscale Resolution in Separated Unsteady High Speed Turbulent Flows

2006 ◽  
Author(s):  
D. Basu ◽  
A. Hamed ◽  
K. Das
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Turbulent Flows ◽  
Sound Pressure Level ◽  
High Speed ◽  
Sound Pressure ◽  
Pressure Fluctuations ◽  
Pressure Level ◽  
Navier Stokes ◽  
Detached Eddy Simulation

This study deals with the computational grid requirements in multiscale simulations of separated turbulent flows at high Reynolds number. The two-equation k-ε based DES (Detached Eddy Simulation) model is implemented in a full 3-D Navier-Stokes solver and numerical results are presented for transonic flow solution over an open cavity. Results for the vorticity, pressure fluctuations, SPL (Sound Pressure level) spectra and for modeled and resolved TKE (Turbulent Kinetic Energy) are presented and compared with available experimental data and with LES results. The results indicate that grid resolution significantly influences the resolved scales and the peak amplitude of the unsteady sound pressure level (SPL) and turbulent kinetic energy spectra.

On the structure of shear stress and turbulent kinetic energy flux across the roughness layer of a gravel-bed channel flow

2009 ◽  
Vol 638 ◽  
pp. 423-452 ◽  
Author(s):  
EMMANUEL MIGNOT ◽  
D. HURTHER ◽  
E. BARTHELEMY
Keyword(s):  
Shear Stress ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Channel Flow ◽  
Mean Flow ◽  
Local Flow ◽  
Gravel Bed ◽  
Wall Flows ◽  
The Mean ◽  
Conditional Statistics

This study examines the structure of shear stress and turbulent kinetic energy (TKE) flux across the roughness layer of a uniform, fully rough gravel-bed channel flow (ks+ ≫ 100, δ/k = 20) using high-resolution acoustic Doppler velocity profiler measurements. The studied gravel-bed roughness layer exhibits a complex random multi-scale roughness structure in strong contrast with conceptualized k- or d-type roughness in standard rough-wall flows. Within the roughness layer, strong spatial variability of all time-averaged flow quantities are observed affecting up to 40% of the boundary layer height. This variability is attributed to the presence of bed zones with emanating bed protuberances (or gravel clusters) acting as local flow obstacles and bed zones of more homogenous roughness of densely packed gravel elements. Considering the strong spatial mean flow variability across the roughness layer, a spatio-temporal averaging procedure, called double averaging (DA), has been applied to the analysed flow quantities. Three aspects have been addressed: (a) the DA shear stress and DA TKE flux in specific bed zones associated with three classes of velocity profiles as previously proposed in Mignot, Barthélemy & Hurther (J. Fluid Mech., vol. 618, 2009, p. 279), (b) the global and per class DA conditional statistics of shear stress and associated TKE flux and (c) the contribution of large-scale coherent shear stress structures (LC3S) to the TKE flux across the roughness layer. The mean Reynolds and dispersive shear structure show good agreement between the protuberance bed zones associated with the S-shape/accelerated classes and recent results obtained in standard k-type rough-wall flows (Djenidi et al., Exp. Fluids, vol. 44, 2008, p. 37; Pokrajac, McEwan & Nikora, Exp. Fluids, vol. 45, 2008, p. 73). These gravel-bed protuberances act as local flow obstacles inducing a strong turbulent activity in their wake regions. The conditional statistics show that the Reynolds stress contribution is fairly well distributed between sweep and ejection events, with threshold values ranging from H = 0 to H = 8. However, the TKE flux across the roughness layer primarily results from the residual shear stress between ejection and sweep of very high magnitude (H = 10–20) and of small turbulent scale. Although LC3S are seen to penetrated the interfacial roughness layer, their TKE flux contribution is found to be negligible compared to the very energetic small-scale sweep events. These sweeps are dominantly produced in the bed zones of local gravel protuberances where the velocity profiles are inflexional of S-shape type and the mean flow properties are of mixing-layer flow type as previously shown in Mignot et al. (2009).

On the interactions between large-scale structure and fine-grained turbulence in a free shear flow I. The development of temporal interactions in the mean

1976 ◽  
Vol 352 (1669) ◽  
pp. 213-247 ◽  
Keyword(s):  
Energy Transfer ◽  
Kinetic Energy ◽  
Shear Layer ◽  
Turbulent Kinetic Energy ◽  
Viscous Dissipation ◽  
Mean Flow ◽  
Fine Grained ◽  
Large Eddy ◽  
The Mean ◽  
Three Components

In this problem a mean turbulent shear layer originally exists, homogeneous in the streamwise direction, formed perhaps by previous instabilities, but in equilibrium with the fine-grained turbulence. At a given time, a large eddy of a fixed horizontal wavenumber is initiated. We study the subsequent time development of the non-equilibrium interactions between the three components of flow as they adjust towards ultimate simultaneous equilibrium, using the integrated energy-balance conservation equations to derive the amplitude equations. This necessarily involves the usual averaging procedure and a conditional or phase-averaging procedure by which the large structure motion is educed from the total fluctuations. In general, the mean flow growth is due to the energy transfer to both fluctuating components, the large eddy gains energy from the mean motion and exchanges energy with the fine-grained turbulence, while the fine-grained turbulence gains energy from the mean flow and exchanges with the large eddy and converts its energy to heat through viscous dissipation of the smallest scales. The closure problem is obtained via the shape assumptions which enter into the interaction integrals. The situation in which the fine-grained turbulent kinetic energy production and viscous dissipation are in local balance is considered, the displacement from equilibrium being due only to the energy transfer from the large eddy. The large eddy shape is taken to be two-dimensional, instability-wavelike, with its vorticity axis perpendicular to the direction of the mean outer stream. Prior to averaging, detailed but approximate calculations of the wave-induced turbulent Reynolds stresses are obtained; the product of these stresses with the appropriate large-eddy rates of strain give the energy transfer mechanism between the two disparate scales of fluctuations. Coupled, nonlinear amplitude or energy density equations for the three components of motion are obtained, the coefficients of which are the interaction integrals guided by the shape assumptions. It is found that for the special case of parallel flow, the energy of the large eddy first undergoes a hydrodynamic-instability type of amplification but eventually decays due to the energy transfer to the fine-grained turbulence, while the turbulent kinetic energy is displaced from an original level of equilibrium to a new one because of the ability of the large eddy to negotiate an indirect energy transfer from the mean flow. For the growing shear layer, approximate considerations show that if the mechanism of energy transfer from the large to the small scale is eventually weakened by the shear layer growth compared to the large-eddy production mechanism so that the amplification and decay process repeats, ‘bursts’ of the remnant of the same large eddy will occur repeatedly until an ultimate equilibrium is reached among the three interacting components of motion. However, for the large eddy whose wavenumber corresponds to that of the initially most amplified case, the ‘bursting’ phenomenon is much less pronounced and equilibrium is very nearly reached at the end of the very first ‘burst’.

The Structure of Forward Facing Step Flows in Adverse Pressure Gradient

2014 ◽  
Author(s):  
Weijie Shao ◽  
Martin Agelin-Chaab
Keyword(s):  
Pressure Gradient ◽  
Turbulent Flows ◽  
Decomposition Analysis ◽  
Adverse Pressure Gradient ◽  
Quality Data ◽  
Reattachment Length ◽  
Mean Flow ◽  
Particle Image Velocimetry Technique ◽  
The Mean ◽  
Forward Facing Step

This paper reports an investigation of the effects of adverse pressure gradient on turbulent flows over forward facing step. Three adverse pressure gradients were created for this study using diverging channels. A particle image velocimetry technique was used to conduct measurements in the streamwise-wall-normal (x-y) planes at the mid-plane of test section at several locations downstream to 68 step heights. A Reynolds number of Reh = 4800 and δ/h = 4.7 were employed, where h is the mean step height and δ is the approach boundary layer thickness. The results include the mean flow and turbulence quantities as well as proper orthogonal decomposition analysis. The mean reattachment length obtained indicates that the adverse pressure gradient created in this study does not have significant effects on the reattachment length. The triple velocity correlations imply that there is negative transport of turbulence kinetic energy close to the wall and positive transport away from the wall. In addition to the physical insight, the high quality data reported are useful for assessing the ability of turbulence models to reproduce the behaviour of complex flows.

scholarly journals The streamwise turbulence intensity in the intermediate layer of turbulent pipe flow

2015 ◽  
Vol 774 ◽  
pp. 324-341 ◽  
Author(s):  
J. C. Vassilicos ◽  
J.-P. Laval ◽  
J.-M. Foucaut ◽  
M. Stanislas
Keyword(s):  
Reynolds Number ◽  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Intermediate Layer ◽  
High Reynolds Number ◽  
Logarithmic Derivative ◽  
Turbulent Pipe Flow ◽  
Mean Flow ◽  
The Mean ◽  
High Reynolds

The spectral model of Perryet al. (J. Fluid Mech., vol. 165, 1986, pp. 163–199) predicts that the integral length scale varies very slowly with distance to the wall in the intermediate layer. The only way for the integral length scale’s variation to be more realistic while keeping with the Townsend–Perry attached eddy spectrum is to add a new wavenumber range to the model at wavenumbers smaller than that spectrum. This necessary addition can also account for the high-Reynolds-number outer peak of the turbulent kinetic energy in the intermediate layer. An analytic expression is obtained for this outer peak in agreement with extremely high-Reynolds-number data by Hultmarket al. (Phys. Rev. Lett., vol. 108, 2012, 094501;J. Fluid Mech., vol. 728, 2013, pp. 376–395). Townsend’s (The Structure of Turbulent Shear Flows, 1976, Cambridge University Press) production–dissipation balance and the finding of Dallaset al. (Phys. Rev. E, vol. 80, 2009, 046306) that, in the intermediate layer, the eddy turnover time scales with skin friction velocity and distance to the wall implies that the logarithmic derivative of the mean flow has an outer peak at the same location as the turbulent kinetic energy. This is seen in the data of Hultmarket al. (Phys. Rev. Lett., vol. 108, 2012, 094501;J. Fluid Mech., vol. 728, 2013, pp. 376–395). The same approach also predicts that the logarithmic derivative of the mean flow has a logarithmic decay at distances to the wall larger than the position of the outer peak. This qualitative prediction is also supported by the aforementioned data.

PIV Study of Adverse and Favorable Pressure Gradient Turbulent Flows Over Transverse Ribs

2008 ◽  
Vol 130 (11) ◽  
Author(s):  
M. Agelinchaab ◽  
M. F. Tachie
Keyword(s):  
Pressure Gradient ◽  
Turbulent Flows ◽  
Reynolds Shear Stress ◽  
Adverse Pressure Gradient ◽  
Bottom Wall ◽  
Combined Effects ◽  
Mean Flow ◽  
Turbulent Statistics ◽  
Favorable Pressure Gradient ◽  
The Mean

This paper reports an experimental study of the combined effects of rib roughness and pressure gradient on turbulent flows produced in asymmetric converging and diverging channels. Transverse square ribs with pitch-to-height ratio of 4 were attached to the bottom wall of the channel to produce the rib roughness. A particle image velocimetry technique was used to conduct measurements at several streamwise-transverse planes located upstream, within, and downstream of the converging and diverging sections of the channel. From these measurements, the mean velocities and turbulent statistics at the top plane of the ribs and across the channel were obtained. The data revealed non-negligible wall-normal motion and interaction between the cavities and overlying boundary layers. The different drag characteristics of the rough bottom wall and the smooth top wall produced asymmetric distributions of mean velocity and turbulent statistics across the channel. The asymmetry of these profiles is most extreme in the presence of adverse pressure gradient. Because of the manner in which pressure gradient modifies the mean flow and turbulence production, it was found that the streamwise turbulence intensity and Reynolds shear stress in the vicinity of the ribs are lower in the adverse pressure gradient than in the favorable pressure gradient channel. The results show also that the combined effects of rib roughness and adverse pressure gradient on the turbulent intensity statistics are significantly higher than when roughness and adverse pressure gradient are applied in isolation.

Effect of Impeller Speed Perturbation in a Rushton Impeller Stirred Tank

2012 ◽  
Vol 134 (6) ◽  
Author(s):  
Somnath Roy ◽  
Sumanta Acharya
Keyword(s):  
Kinetic Energy ◽  
Turbulent Kinetic Energy ◽  
Rotational Speed ◽  
Stirred Tank ◽  
Azimuthal Direction ◽  
Mean Flow ◽  
Eddy Simulation ◽  
Large Eddy ◽  
The Mean ◽  
Rushton Impeller

Flow inside an unbaffled Rushton-impeller stirred tank reactor (STR) is perturbed using a time dependent impeller rotational speed. Large eddy simulation (LES) revealed that the perturbation increased the width of impeller jet compared to the constant rotational speed cases. The turbulent fluctuations were also observed to be enhanced in the perturbed flow and showed higher values of production and convection of turbulent kinetic energy. Changes in the mean flow-field during the perturbation cycle are investigated. The trailing edge vortices were observed to propagate farther both in the radial and azimuthal direction in the perturbed case. Production of turbulent kinetic energy is observed to be related to the breakup of the impeller jet in the perturbed case. Dissipation of turbulent kinetic energy is augmented due to the perturbation ensuring a better mixing at the molecular scale.

Turbulent boundary layers absent mean shear

2017 ◽  
Vol 835 ◽  
pp. 217-251 ◽  
Author(s):  
Blair A. Johnson ◽  
Edwin A. Cowen
Keyword(s):  
Boundary Layer ◽  
Kinetic Energy ◽  
Turbulent Boundary Layer ◽  
Turbulent Kinetic Energy ◽  
Temporal Frequency ◽  
Synthetic Jet ◽  
Mean Flow ◽  
Frequency Spectra ◽  
Empirical Constant ◽  
The Mean

We perform an experimental study to investigate the turbulent boundary layer above a stationary solid glass bed in the absence of mean shear. High Reynolds number $(Re_{\unicode[STIX]{x1D706}}\sim 300)$ horizontally homogeneous isotropic turbulence is generated via randomly actuated synthetic jet arrays (RASJA – Variano & Cowen J. Fluid Mech. vol. 604, 2008, pp. 1–32). Each of the arrays is controlled by a spatio-temporally varying algorithm, which in turn minimizes the formation of secondary mean flows. One array consists of an $8\times 8$ grid of jets, while the other is a $16\times 16$ array. Particle image velocimetry measurements are used to study the isotropic turbulent region and the boundary layer formed beneath as the turbulence encounters a stationary wall. The flow is characterized with statistical metrics including the mean flow and turbulent velocities, turbulent kinetic energy, integral scales and the turbulent kinetic energy transport equation, which includes the energy dissipation rate, production and turbulent transport. The empirical constant in the Tennekes (J. Fluid Mech. vol. 67, 1975, pp. 561–567) model of Eulerian frequency spectra is calculated based on the dissipation results and temporal frequency spectra from acoustic Doppler velocimetry measurements. We compare our results to prior literature that addresses mean shear free turbulent boundary layer characterizations via grid-stirred tank experiments, moving-bed experiments, rapid-distortion theory and direct numerical simulations in a forced turbulent box. By varying the operational parameters of the randomly actuated synthetic jet array, we also find that we are able to control the turbulence levels, including integral length scales and dissipation rates, by changing the mean on-times in the jet algorithm.

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