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.

Studying turbulence using direct numerical simulation: 1987 Center for Turbulence Research NASA Ames/Stanford Summer Programme

1988 ◽  
Vol 190 ◽  
pp. 375-392 ◽  
Author(s):  
J. C. R. Hunt
Keyword(s):  
Boundary Layers ◽  
Turbulent Flows ◽  
Large Scale ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Turbulent Boundary Layers ◽  
Finite Amplitude ◽  
Wall Layer ◽  
Navier Stokes ◽  
Scale Motion

This paper is an account of a summer programme for the study of the ideas and models of turbulent flows, using the results of direct numerical stimulations of the Navier-Stokes equations. These results had been obtained on the computers and stored as accessible databases at the Center for Turbulence Research (CTR) of NASA Ames Research Center and Stanford University. At this first summer programme, some 32 visiting researchers joined those at the CTR to test hypotheses and models in five aspects of turbulence research: turbulence decomposition, bifurcation and chaos; two-point closure (or k-space) modelling; structure of turbulent boundary layers; Reynolds-stress modelling; scalar transport and reacting flows.A number of new results emerged including: computation of space and space-time correlations in isotropic turbulence can be related to each other and modelled in terms of the advection of small scales by large-scale motion; the wall layer in turbulent boundary layers is dominated by shear layers which protrude into the outer layers, and have long lifetimes; some aspects of the ejection mechanism for these layers can be described in terms of the two-dimensional finite-amplitude Navier-Stokes solutions; a self-similar form of the two-point, cross-correlation data of the turbulence in boundary layers (when normalized by the r.m.s. value at the furthest point from the wall) shows how both the blocking of eddies by the wall and straining by the mean shear control the lengthscales; the intercomponent transfer (pressure-strain) is highly localized in space, usually in regions of concentrated vorticity; conditioned pressure gradients are linear in the conditioning of velocity and independent of vorticity in homogeneous shear flow; some features of coherent structures in the boundary layer are similar to experimental measurements of structures in mixing-layers, jets and wakes.The availability of comprehensive velocity and pressure data certainly helps the investigation of concepts and models. But a striking feature of the summer programme was the diversity of interpretation of the same computed velocity fields. There are few signs of any convergence in turbulence research! But with new computational facilities the divergent approaches can at least be related to each other.

Turbulent Flows

1997 ◽  
Author(s):  
David Jon Furbish
Keyword(s):  
Turbulent Flows ◽  
High Speed ◽  
Stokes Equations ◽  
Fluid Motion ◽  
Navier Stokes ◽  
River Channels ◽  
Navier Stokes Equations ◽  
Viscous Forces ◽  
Thermally Driven ◽  
Flow Phenomena

Many geological flows involve turbulence, wherein the velocity field involves complex, fluctuating motions superimposed on a mean motion. Flows in natural river channels are virtually always turbulent. Magma flow in dikes and sills, and lava flows, can be turbulent. Atmospheric flows involving eolian transport are turbulent. The complex, convective overturning of fluid in a magma chamber or geyser is a form of turbulence. Thus, a description of the basic qualities of these complex flows is essential for understanding many geological flow phenomena. Turbulent flows generally are associated with large Reynolds numbers. Recall from Chapter 5 that the Reynolds number Re is a measure of the ratio of inertial to viscous forces acting on a fluid element, . . . Re = ρUL/μ . . . . . . (14.1) . . . where the characteristic velocity U and length L are defined in terms of the particular flow system. Thus, turbulence is typically associated, for given fluid density ρ and viscosity μ, with high-speed flows (although we must be careful in applying this generality to thermally driven convective motions; see Chapter 16). A simple, visual illustration of this occurs when smoke rises from a cigar within otherwise calm, surrounding air. The smoke acts as a flow tracer. Smoke molecules at the cigar tip start from rest, since they are initially attached to the cigar. Upward fluid motion, as traced by the smoke, initially is of low speed, and viscous forces have a relatively important influence on its behavior. The flow is laminar; smoke streaklines are smooth and locally parallel. But as the flow accelerates upward, it typically reaches a point where viscous forces are no longer sufficient to damp out destabilizing effects of growing inertial forces, and the flow becomes turbulent, manifest as whirling, swirling fluid motions (see Tolkien [1937]). Throughout this chapter we will consider only incompressible Newtonian fluids. Unfortunately, the complexity of turbulent fluid motions precludes directly using the Navier–Stokes equations to describe them. Instead, we will adopt a procedure whereby the Navier–Stokes equations are recast in terms of temporally averaged or spatially averaged values of velocity and pressure, and fluctuations about these averages.

A Markovian random coupling model for turbulence

1974 ◽  
Vol 65 (1) ◽  
pp. 145-152 ◽  
Author(s):  
U. Frisch ◽  
M. Lesieur ◽  
A. Brissaud
Keyword(s):  
Stochastic Model ◽  
Master Equation ◽  
Turbulent Flows ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Planck Equation ◽  
Navier Stokes ◽  
Coupling Coefficients ◽  
Navier Stokes Equations ◽  
Random Coupling

The Markovian random coupling (MRC) model is a modified form of the stochastic model of the Navier-Stokes equations introduced by Kraichnan (1958, 1961). Instead of constant random coupling coefficients, white-noise time dependence is assumed for the MRC model. Like the Kraichnan model, the MRC model preserves many structural properties of the original Navier-Stokes equations and should be useful for investigating qualitative features of turbulent flows, in particular in the limit of vanishing viscosity. The closure problem is solved exactly for the MRC model by a technique which, contrary to the original Kraichnan derivation, is not based on diagrammatic expansions. A closed equation is obtained for the functional probability distribution of the velocity field which is a special case of Edwards’ (1964) Fokker-Planck equation; this equation is an exact consequence of the stochastic model whereas Edwards’ equation constitutes only the first step in a formal expansion based directly on the Navier-Stokes equations. From the functional equation an exact master equation is derived for simultaneous second-order moments which happens to be essentially a Markovianized version of the single-time quasi-normal approximation characterized by a constant triad-interaction time.The explicit form of the MRC master equation is given for the Burgers equation and for two- and three-dimensional homogeneous isotropic turbulence.

The role of bulk viscosity on the decay of compressible, homogeneous, isotropic turbulence

2017 ◽  
Vol 833 ◽  
pp. 717-744 ◽  
Author(s):  
Shaowu Pan ◽  
Eric Johnsen
Keyword(s):  
Energy Transfer ◽  
Turbulent Flows ◽  
Bulk Viscosity ◽  
Shear Viscosity ◽  
Isotropic Turbulence ◽  
Stokes Equations ◽  
Compressible Turbulence ◽  
Homogeneous Isotropic Turbulence ◽  
Viscous Effects ◽  
Bulk Viscous

While Stokes’ hypothesis of neglecting bulk viscous effects is exact for monatomic gases and unlikely to strongly affect the dynamics of fluids whose bulk-to-shear viscosity ratio is small and/or of weakly compressible turbulence, it is unclear to what extent this assumption holds for compressible, turbulent flows of gases whose bulk viscosity is orders of magnitude larger than their shear viscosities (e.g. $\text{CO}_{2}$). Our objective is to understand the mechanisms by which bulk viscosity and the associated phenomena affect moderately compressible turbulence, in particular energy transfer and dissipation. Using direct numerical simulations of the compressible Navier–Stokes equations, we study the decay of compressible, homogeneous, isotropic turbulence for ratios of bulk-to-shear viscosity ranging from 0 to 1000. Our simulations demonstrate that bulk viscosity increases the decay rate of turbulent kinetic energy; whereas enstrophy exhibits little sensitivity to bulk viscosity, dilatation is reduced by over two orders of magnitude within the first two eddy-turnover times. Via a Helmholtz decomposition of the flow, we determine that the primary action of bulk viscosity is to damp the dilatational velocity fluctuations and reduce dilatational–solenoidal exchanges, as well as pressure–dilatation coupling. In short, bulk viscosity renders compressible turbulence incompressible by reducing energy transfer between translational and internal degrees of freedom. Our results indicate that for gases whose bulk viscosity is of the order of their shear viscosity (e.g. hydrogen) the turbulence is not significantly affected by bulk viscous dissipation, in which case neglecting bulk viscosity is acceptable in practice. However, in problems involving compressible, turbulent flows of gases like $\text{CO}_{2}$ whose bulk viscosities are thousands of times greater than their shear viscosities, bulk viscosity cannot be ignored.

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.

scholarly journals Numerical simulation of flow-induced acoustic oscillations around circular cylinders

2020 ◽  
Vol 168 ◽  
pp. 00055
Author(s):  
Serhii Mirnyi ◽  
Oleg Polevoy ◽  
Andrii Zinchenko ◽  
Anton Pylypenko ◽  
Vasyl Vlasenko
Keyword(s):  
Numerical Simulation ◽  
Turbulent Flows ◽  
Incompressible Flows ◽  
Stokes Equations ◽  
Mining Industry ◽  
Laminar Flows ◽  
Circular Cylinders ◽  
Acoustic Oscillations ◽  
Acoustic Analogy ◽  
Differential Model

Questions of numerical simulation of acoustic oscillations generation modes in the liquid flow around the groups of two and three circular cylinders are considered. In mining industry the processes of hydrodynamic impact on gas-saturated porous media produce significant acoustic emission both at the injection stage and at the liquid discharge stage. Simulation of such kind of acoustic processes is one of the actual problems of theoretical and applied fluid mechanics and under certain assumptions could be reduced to the flow around a group of bodies. Two approaches for numerical simulation of the acoustic oscillations generation induced by the flow around circular cylinders based on numerical solution of the Navier-Stokes equations for compressible and incompressible flows closed by differential model of turbulence and complemented by acoustic analogy equations have been developed. For laminar flows, eight different modes that fundamentally differ both in the flow structure and in the frequency spectrum of parameter oscillations have been identified. For turbulent flows, the classification criteria for the three main frequency modes are presented. Acoustic data are obtained using the Direct Noise Computation technology and acoustic analogies as well.

scholarly journals The flow and flow-induced noise behaviour of a simplified high-speed train bogie in the cavity with and without a fairing

2017 ◽  
Vol 232 (3) ◽  
pp. 759-773 ◽  
Author(s):  
JY Zhu ◽  
ZW Hu ◽  
DJ Thompson
Keyword(s):  
High Speed ◽  
Stokes Equations ◽  
Current Model ◽  
Outer Region ◽  
Surface Shape ◽  
Aerodynamic Noise ◽  
High Speed Train ◽  
Detached Eddy Simulation ◽  
Acoustic Analogy ◽  
Flow Induced Noise

Aerodynamic noise is a significant source for high-speed trains but its prediction in an industrial context is difficult to achieve. In this article, the flow and aerodynamic noise behaviour of a simplified high-speed train bogie at scale 1:10 are studied through numerical simulations. The bogie is situated in a cavity beneath the train and the influence of a bogie fairing on the flow and flow-induced noise that developed around the bogie area is investigated. A two-stage hybrid method is used, which combines the computational fluid dynamics and an acoustic analogy. The near-field unsteady flow is obtained by solving the unsteady three-dimensional Navier–Stokes equations numerically using delayed detached-eddy simulation, and the data are utilised to predict the far-field noise based on the Ffowcs Williams–Hawkings acoustic analogy. Results show that when the bogie is located inside the bogie cavity, the shear layer developed from the leading edges of the cavity interacts strongly with the flow separated from the upstream components of the bogie and the cavity walls. Therefore, a highly turbulent flow is generated within the bogie cavity due to the strong flow impingements and flow recirculations occurring there. For the case without the fairing, the surface shape discontinuity in the bogie cavity along the carbody sidewalls generates strong flow unsteadiness around these regions. When the fairing is mounted in front of the bogie cavity, the flow interactions between the bogie cavity and the outer region are reduced and the development of turbulence outside the fairing is greatly weakened. Based on the predictions of the noise radiated to the trackside using a permeable data surface parallel to the carbody sidewall, it has been found that the bogie fairing is effective in reducing the noise generated in most of the frequency range, and a noise reduction of around 5 dB is achieved in the farfield for the current model case.

scholarly journals Redchyts Evaluation of aerodynamic and thermal loads on the HYPERLOOP capsule fuselage in a partly evacuated tube

2019 ◽  
Vol 4 (123) ◽  
pp. 3-12
Author(s):  
Oleh Borysovych Polovyi ◽  
Dmytro Oleksandrovych Redchyts
Keyword(s):  
High Speed ◽  
Stokes Equations ◽  
Aerodynamic Drag ◽  
Negative Impact ◽  
Transportation Systems ◽  
Transport Systems ◽  
Aerodynamic Heating ◽  
Thermal Loads ◽  
Important Place ◽  
Evacuated Tube

Aerodynamics occupies an important place in the design of high-speed ground transportation systems. When a vehicle is moving at a speed above 500 km/h under atmospheric pressure, the main energy is spent to overcome the aerodynamic drag. Creating a rarefied atmosphere inside a sealed pipe in order to significantly reduce energy loss is one of the key ideas of the HYPERLOOP project [1].The paper assesses the aerodynamic and thermal loads on the HYPERLOOP capsule fuselage in a partly evacuated tube based on the numerical solution of the Navier-Stokes equations of compressible flow closed by a differential turbulence model [2-4]. Numerical modeling was carried out with the help of the computational fluid dynamics software developed by the scientific researchers of the Institute of Transport Systems and Technologies of the National Academy of Sciences of Ukraine [5].It was shown that even under conditions of low air pressure in a partly evacuated tube the high-speed movement of the HYPERLOOP capsule will be accompanied by the formation of local supersonic zones, shock waves and non-stationary vortex systems. The structure of the flow essentially depends on geometry of the streamlined capsule and the speed of its movement.It was found that the flow structure and the values of aerodynamic dimensionless coefficients weakly depend on the pressure in the partly evacuated tube. Thus, the aerodynamic forces acting on the HYPERLOOP capsule at the same speeds are almost directly proportional to the pressure value in the tube.A certain problem in the design of the HYPERLOOP type high-speed vehicles will be the aerodynamic heating of the capsule fuselage. When the capsule moves at transonic speed the temperature of the outer surface of the capsule will be 60÷900 C. This heat load can have a negative impact on the performance of onboard power supply and control systems, as well as on the ensuring of the passengers’ comfort on the way.

scholarly journals Finite-scale equations for compressible fluid flow

2009 ◽  
Vol 367 (1899) ◽  
pp. 2861-2871 ◽  
Author(s):  
L.G. Margolin
Keyword(s):  
Numerical Simulation ◽  
Turbulent Flows ◽  
Compressible Fluid ◽  
High Speed ◽  
Energy Transport ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
One Dimensional ◽  
Scale Models

Finite-scale equations (FSE) describe the evolution of finite volumes of fluid over time. We discuss the FSE for a one-dimensional compressible fluid, whose every point is governed by the Navier–Stokes equations. The FSE contain new momentum and internal energy transport terms. These are similar to terms added in numerical simulation for high-speed flows (e.g. artificial viscosity) and for turbulent flows (e.g. subgrid scale models). These similarities suggest that the FSE may provide new insight as a basis for computational fluid dynamics. Our analysis of the FS continuity equation leads to a physical interpretation of the new transport terms, and indicates the need to carefully distinguish between volume-averaged and mass-averaged velocities in numerical simulation. We make preliminary connections to the other recent work reformulating Navier–Stokes equations.

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.

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