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A. Patel
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This paper presents a novel methodology for improving eddy viscosity models in predicting wall-bounded turbulent flows with strong gradients in the thermo-physical properties. Common turbulence models for solving the Reynolds-averaged Navier–Stokes equations do not correctly account for variations in transport properties, such as density and viscosity, which can cause substantial inaccuracies in predicting important quantities of interest, for example, heat transfer and drag. Based on the semi-locally scaled turbulent kinetic energy equation, introduced in [Pecnik and Patel, J. Fluid Mech. (2017), vol. 823, R1], we analytically derive a modification of the diffusion term of turbulent scalar equations. The modification has been applied to five common eddy viscosity turbulence models and tested for fully developed turbulent channels with isothermal walls that are volumetrically heated, either by a uniform heat source or viscous heating in supersonic flow conditions. The agreement with results obtained by direct numerical simulation shows that the modification significantly improves results of eddy viscosity models for fluids with variable transport properties.
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This paper presents a novel methodology for improving eddy viscosity models in predicting wall-bounded turbulent flows with strong gradients in the thermo-physical properties. Common turbulence models for solving the Reynolds-averaged Navier–Stokes equations do not correctly account for variations in transport properties, such as density and viscosity, which can cause substantial inaccuracies in predicting important quantities of interest, for example, heat transfer and drag. Based on the semi-locally scaled turbulent kinetic energy equation, introduced in [Pecnik and Patel, J. Fluid Mech. (2017), vol. 823, R1], we analytically derive a modification of the diffusion term of turbulent scalar equations. The modification has been applied to five common eddy viscosity turbulence models and tested for fully developed turbulent channels with isothermal walls that are volumetrically heated, either by a uniform heat source or viscous heating in supersonic flow conditions. The agreement with results obtained by direct numerical simulation shows that the modification significantly improves results of eddy viscosity models for fluids with variable transport properties.
The present work consists of an investigation of the turbulence radiation interaction (TRI) in a radiative turbulent channel flow of grey gas bounded by isothermal hot and cold walls. The optical thickness of the channel is varied from 0.1 to 10 to observe different regimes of TRI. A high-resolution finite volume method for radiative heat transfer is employed and coupled with the direct numerical simulation (DNS) of the flow. The resulting effects of TRI on temperature statistics are strongly dependent on the considered optical depth. In particular, the contrasting role of emission and absorption is highlighted. For a low optical thickness the effect of radiative fluctuations on temperature statistics is low and causes the reduction of temperature variance through the dissipating action of emission. On the other hand, while increasing optical thickness to relatively high levels, the dissipation of temperature variance is balanced, at low wavenumbers in the turbulence spectrum, through the preferential action of absorption, which increases the large-scale temperature fluctuations. A significant rise in the effect of radiation on the temperature variance can be observed as a consequence of the reduction of radiative heat transfer length scales.
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The present work consists of an investigation of the turbulence radiation interaction (TRI) in a radiative turbulent channel flow of grey gas bounded by isothermal hot and cold walls. The optical thickness of the channel is varied from 0.1 to 10 to observe different regimes of TRI. A high-resolution finite volume method for radiative heat transfer is employed and coupled with the direct numerical simulation (DNS) of the flow. The resulting effects of TRI on temperature statistics are strongly dependent on the considered optical depth. In particular, the contrasting role of emission and absorption is highlighted. For a low optical thickness the effect of radiative fluctuations on temperature statistics is low and causes the reduction of temperature variance through the dissipating action of emission. On the other hand, while increasing optical thickness to relatively high levels, the dissipation of temperature variance is balanced, at low wavenumbers in the turbulence spectrum, through the preferential action of absorption, which increases the large-scale temperature fluctuations. A significant rise in the effect of radiation on the temperature variance can be observed as a consequence of the reduction of radiative heat transfer length scales.
Book chapter
(2017)
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Ashish Patel, Rene Pecnik, Jurriaan Peeters, Bendiks Jan Boersma, Stefan Hickel, M.E. Moghadam
We investigate the effectiveness of the semi-local Reynolds number Re τ to parametrize wall-bounded flows with strong density, ρ, and viscosity, µ, gradients. Several cases are considered, namely, volumetrically heated low-Mach-number turbulent channel flows, a simultaneously heated and cooled flow with CO2 at supercritical pressure, and heated and cooled supersonic boundary layer flows. The mean density and viscosity in some of these cases vary up to a factor of nine and six, respectively. We show that, even for such high gradients in mean properties, the velocity transformation based on the semi-local Reynolds number is able to collapse the mean streamwise velocity profiles. We further-more provide evidence that the turbulent kinetic energy and streamwise vorticity budget equations are also governed by the semi-local Reynolds number. For cases with strong property variations, additional mechanisms appear that are caused by individual density (e.g., baroclinicity) or viscosity gradients. However, in the cases investigated herein, these additional mechanisms are small. The insights gained are used to improve a wall model, which is then tested in a wall-modeled large-eddy simulation (LES) of a compressible channel flow with isothermal walls.
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We investigate the effectiveness of the semi-local Reynolds number Re τ to parametrize wall-bounded flows with strong density, ρ, and viscosity, µ, gradients. Several cases are considered, namely, volumetrically heated low-Mach-number turbulent channel flows, a simultaneously heated and cooled flow with CO2 at supercritical pressure, and heated and cooled supersonic boundary layer flows. The mean density and viscosity in some of these cases vary up to a factor of nine and six, respectively. We show that, even for such high gradients in mean properties, the velocity transformation based on the semi-local Reynolds number is able to collapse the mean streamwise velocity profiles. We further-more provide evidence that the turbulent kinetic energy and streamwise vorticity budget equations are also governed by the semi-local Reynolds number. For cases with strong property variations, additional mechanisms appear that are caused by individual density (e.g., baroclinicity) or viscosity gradients. However, in the cases investigated herein, these additional mechanisms are small. The insights gained are used to improve a wall model, which is then tested in a wall-modeled large-eddy simulation (LES) of a compressible channel flow with isothermal walls.
We investigate the stability of streaks in the buffer layer of turbulent channel flows with temperature-dependent density and viscosity by means of linear theory. The adopted framework consists of an extended set of the Orr-Sommerfeld-Squire equations that accounts for density and viscosity nonuniformity in the direction normal to the walls. The base flow profiles for density, viscosity, and velocity are averaged from direct numerical simulations (DNSs) of fully developed turbulent channel flows. We find that the inner scaling based on semilocal quantities provides an effective parametrization of the effect of variable properties on the linearized flow. The spanwise spacing of optimal buffer layer streaks scales to λz,opt≈90 for all cases considered and the maximum energy amplification decreases, compared to the one for a flow with constant properties, if the semilocal Reynolds number Reτ increases away from the walls, consistently with less energetic streaks observed in DNSs of turbulent channels. A secondary stability analysis of the two-dimensional velocity profile formed by the mean turbulent velocity and the nonlinearly saturated optimal streaks predicts a streamwise instability mode with wavelength λx,cr≈230 in semilocal units, regardless of the fluid property distribution across the channel. The threshold for the onset of the secondary instability is reduced, compared to a constant property flow, if Reτ increases away from the walls, which explains the more intense ejection events reported in DNSs. The opposite behavior is predicted by the linear theory for decreasing Reτ, in accord with DNS observations. We finally show that the phase velocity of the critical mode of secondary instability agrees well with the convection velocity calculated by DNSs in the near-wall region for both constant and variable viscosity flows.
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We investigate the stability of streaks in the buffer layer of turbulent channel flows with temperature-dependent density and viscosity by means of linear theory. The adopted framework consists of an extended set of the Orr-Sommerfeld-Squire equations that accounts for density and viscosity nonuniformity in the direction normal to the walls. The base flow profiles for density, viscosity, and velocity are averaged from direct numerical simulations (DNSs) of fully developed turbulent channel flows. We find that the inner scaling based on semilocal quantities provides an effective parametrization of the effect of variable properties on the linearized flow. The spanwise spacing of optimal buffer layer streaks scales to λz,opt≈90 for all cases considered and the maximum energy amplification decreases, compared to the one for a flow with constant properties, if the semilocal Reynolds number Reτ increases away from the walls, consistently with less energetic streaks observed in DNSs of turbulent channels. A secondary stability analysis of the two-dimensional velocity profile formed by the mean turbulent velocity and the nonlinearly saturated optimal streaks predicts a streamwise instability mode with wavelength λx,cr≈230 in semilocal units, regardless of the fluid property distribution across the channel. The threshold for the onset of the secondary instability is reduced, compared to a constant property flow, if Reτ increases away from the walls, which explains the more intense ejection events reported in DNSs. The opposite behavior is predicted by the linear theory for decreasing Reτ, in accord with DNS observations. We finally show that the phase velocity of the critical mode of secondary instability agrees well with the convection velocity calculated by DNSs in the near-wall region for both constant and variable viscosity flows.
We derive an alternative formulation of the turbulent kinetic energy equation for flows with strong near-wall density and viscosity gradients. The derivation is based on a scaling transformation of the Navier-Stokes equations using semi-local quantities. A budget analysis of the semi-locally scaled turbulent kinetic energy equation shows that, for several variable property low-Mach-number channel flows, the 'leading-order effect' of variable density and viscosity on turbulence in wall bounded flows can effectively be characterized by the semi-local Reynolds number. Moreover, if a turbulence model is solved in its semi-locally scaled form, we show that an excellent agreement with direct numerical simulations is obtained for both low- and high-Mach-number flows, where conventional modelling approaches fail.
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We derive an alternative formulation of the turbulent kinetic energy equation for flows with strong near-wall density and viscosity gradients. The derivation is based on a scaling transformation of the Navier-Stokes equations using semi-local quantities. A budget analysis of the semi-locally scaled turbulent kinetic energy equation shows that, for several variable property low-Mach-number channel flows, the 'leading-order effect' of variable density and viscosity on turbulence in wall bounded flows can effectively be characterized by the semi-local Reynolds number. Moreover, if a turbulence model is solved in its semi-locally scaled form, we show that an excellent agreement with direct numerical simulations is obtained for both low- and high-Mach-number flows, where conventional modelling approaches fail.
Direct numerical simulation of fully developed, internally heated channel flows with isothermal walls is performed using the low-Mach-number approximation of Navier-Stokes equation to investigate the influence of temperature-dependent properties on turbulent scalar statistics. Different constitutive relations for density ρ, viscosity μ, and thermal conductivity λ as a function of temperature are prescribed in order to characterize the turbulent scalar statistics. It is shown that the dominant effect caused by property variations on scalar statistics can be parameterized by two nondimensional parameters, namely the semilocal Reynolds number Re★τ≡Reτ√(¯ρ/ρw)/(¯¯μ/μw) (the bar and subscript w denote Reynolds averaging and wall value respectively, while Reτ is the friction Reynolds number based on wall values), and the local Prandtl number Pr★=Prw(¯¯μ/μw)/(¯λ/λw) (Prw is the molecular Prandtl number based on wall values). Near-wall gradients in Re★τ modulate the turbulent heat flux generation mechanism because of structural changes in turbulence. However, the influence of these modulations on the inner scaling of turbulent heat conductivity normalized by local mean viscosity is shown to be weak. Using this observation, a temperature transformation is derived that is invariant of Re★τ variations and only exhibits a Pr★-dependent shift.
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Direct numerical simulation of fully developed, internally heated channel flows with isothermal walls is performed using the low-Mach-number approximation of Navier-Stokes equation to investigate the influence of temperature-dependent properties on turbulent scalar statistics. Different constitutive relations for density ρ, viscosity μ, and thermal conductivity λ as a function of temperature are prescribed in order to characterize the turbulent scalar statistics. It is shown that the dominant effect caused by property variations on scalar statistics can be parameterized by two nondimensional parameters, namely the semilocal Reynolds number Re★τ≡Reτ√(¯ρ/ρw)/(¯¯μ/μw) (the bar and subscript w denote Reynolds averaging and wall value respectively, while Reτ is the friction Reynolds number based on wall values), and the local Prandtl number Pr★=Prw(¯¯μ/μw)/(¯λ/λw) (Prw is the molecular Prandtl number based on wall values). Near-wall gradients in Re★τ modulate the turbulent heat flux generation mechanism because of structural changes in turbulence. However, the influence of these modulations on the inner scaling of turbulent heat conductivity normalized by local mean viscosity is shown to be weak. Using this observation, a temperature transformation is derived that is invariant of Re★τ variations and only exhibits a Pr★-dependent shift.
Wall-bounded turbulence involving mixing of scalars, such as temperature or concentration fields, play an important role in many engineering applications. In applications with large temperature or concentration differences, the variation of scalar dependent thermos physical properties can be strong. In such cases the strong coupling between energy and momentum alters the conventional behavior of turbulence. This alteration results in peculiar momentum and heat transfer characteristics, for which conventional scaling laws for constant property flows fail and cannot be applied. The aim of this work is to characterize wall bounded turbulence for fluids that have large near-wall gradients in thermos physical properties. The focus is on the variable inertia effects at the low-Mach number limit without the influence of buoyancy.
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Wall-bounded turbulence involving mixing of scalars, such as temperature or concentration fields, play an important role in many engineering applications. In applications with large temperature or concentration differences, the variation of scalar dependent thermos physical properties can be strong. In such cases the strong coupling between energy and momentum alters the conventional behavior of turbulence. This alteration results in peculiar momentum and heat transfer characteristics, for which conventional scaling laws for constant property flows fail and cannot be applied. The aim of this work is to characterize wall bounded turbulence for fluids that have large near-wall gradients in thermos physical properties. The focus is on the variable inertia effects at the low-Mach number limit without the influence of buoyancy.
We use direct numerical simulations to study the effect of thermal boundary conditions on developing turbulent pipe flows with fluids at supercritical pressure. The Reynolds number based on pipe diameter and friction velocity at the inlet is Reτ0=360 and Prandtl number at the inlet is Pr0=3.19. The thermodynamic conditions are chosen such that the temperature range within the flow domain incorporates the pseudo-critical point where large variations in thermophysical properties occur. Two different thermal wall boundary conditions are studied: one that permits temperature fluctuations and one that does not allow temperature fluctuations at the wall (equivalent to cases where the thermal effusivity ratio approaches infinity and zero, respectively). Unlike for turbulent flows with constant thermophysical properties and Prandtl numbers above unity – where the effusivity ratio has a negligible influence on heat transfer – supercritical fluids shows a strong dependency on the effusivity ratio. We observe a reduction of 7 % in Nusselt number when the temperature fluctuations at the wall are suppressed. On the other hand, if temperature fluctuations are permitted, large property variations are induced that consequently cause an increase of wall-normal velocity fluctuations very close to the wall and thus an increased overall heat flux and skin friction.
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We use direct numerical simulations to study the effect of thermal boundary conditions on developing turbulent pipe flows with fluids at supercritical pressure. The Reynolds number based on pipe diameter and friction velocity at the inlet is Reτ0=360 and Prandtl number at the inlet is Pr0=3.19. The thermodynamic conditions are chosen such that the temperature range within the flow domain incorporates the pseudo-critical point where large variations in thermophysical properties occur. Two different thermal wall boundary conditions are studied: one that permits temperature fluctuations and one that does not allow temperature fluctuations at the wall (equivalent to cases where the thermal effusivity ratio approaches infinity and zero, respectively). Unlike for turbulent flows with constant thermophysical properties and Prandtl numbers above unity – where the effusivity ratio has a negligible influence on heat transfer – supercritical fluids shows a strong dependency on the effusivity ratio. We observe a reduction of 7 % in Nusselt number when the temperature fluctuations at the wall are suppressed. On the other hand, if temperature fluctuations are permitted, large property variations are induced that consequently cause an increase of wall-normal velocity fluctuations very close to the wall and thus an increased overall heat flux and skin friction.
In this paper we investigate and compare two dierent solar receiver technologies for concentrated solar power plants operating with supercritical CO2. The rst receiver is based on conventional surface absorbers, while the second receiver is based on an innovative idea to use volumetric receivers where sunlight is transmitted through a transparent pipe and directly absorbed by nanoparticles
dispersed in supercritical CO2. The optical properties of the nanoparticles and the CO2 at high pressures and temperatures have been rst estimated and then used in a Navier-Stokes solver that has been coupled to a radiative transfer solver. The results indicate that for temperatures up to 700C the volumetric receiver achieves thermal collector eciencies up to 65% as compared to
surface receivers with 50%.
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In this paper we investigate and compare two dierent solar receiver technologies for concentrated solar power plants operating with supercritical CO2. The rst receiver is based on conventional surface absorbers, while the second receiver is based on an innovative idea to use volumetric receivers where sunlight is transmitted through a transparent pipe and directly absorbed by nanoparticles
dispersed in supercritical CO2. The optical properties of the nanoparticles and the CO2 at high pressures and temperatures have been rst estimated and then used in a Navier-Stokes solver that has been coupled to a radiative transfer solver. The results indicate that for temperatures up to 700C the volumetric receiver achieves thermal collector eciencies up to 65% as compared to
surface receivers with 50%.
The influence of near-wall density and viscosity gradients on near-wall turbulence in a channel is studied by means of direct numerical simulation of the low-Mach-number approximation of the Navier-Stokes equations. Different constitutive relations for density and viscosity as a function of temperature are used in order to mimic a wide range of fluid behaviours and to develop a generalised framework for studying turbulence modulations in variable-property flows. Instead of scaling the velocity solely based on local density, as done for the van Driest transformation, we derive an extension of the scaling that is based on gradients of the semilocal Reynolds number, defined as (the bar and subscript denote Reynolds averaging and wall value respectively, while is the friction Reynolds number based on wall values). This extension of the van Driest transformation is able to collapse velocity profiles for flows with near-wall property gradients as a function of the semilocal wall coordinate. However, flow quantities like mixing length, turbulence anisotropy and turbulent vorticity fluctuations do not show a universal scaling very close to the wall. This is attributed to turbulence modulations, which play a crucial role in the evolution of turbulent structures and turbulence energy transfer. We therefore investigate the characteristics of streamwise velocity streaks and quasistreamwise vortices and find that, similarly to turbulence statistics, the turbulent structures are also strongly governed by profiles and that their dependence on individual density and viscosity profiles is minor. Flows with near-wall gradients in show significant changes in inclination and tilting angles of quasistreamwise vortices. These structural changes are responsible for the observed modulation of the Reynolds stress generation mechanism and the inter-component energy transfer in flows with strong near-wall gradients.
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The influence of near-wall density and viscosity gradients on near-wall turbulence in a channel is studied by means of direct numerical simulation of the low-Mach-number approximation of the Navier-Stokes equations. Different constitutive relations for density and viscosity as a function of temperature are used in order to mimic a wide range of fluid behaviours and to develop a generalised framework for studying turbulence modulations in variable-property flows. Instead of scaling the velocity solely based on local density, as done for the van Driest transformation, we derive an extension of the scaling that is based on gradients of the semilocal Reynolds number, defined as (the bar and subscript denote Reynolds averaging and wall value respectively, while is the friction Reynolds number based on wall values). This extension of the van Driest transformation is able to collapse velocity profiles for flows with near-wall property gradients as a function of the semilocal wall coordinate. However, flow quantities like mixing length, turbulence anisotropy and turbulent vorticity fluctuations do not show a universal scaling very close to the wall. This is attributed to turbulence modulations, which play a crucial role in the evolution of turbulent structures and turbulence energy transfer. We therefore investigate the characteristics of streamwise velocity streaks and quasistreamwise vortices and find that, similarly to turbulence statistics, the turbulent structures are also strongly governed by profiles and that their dependence on individual density and viscosity profiles is minor. Flows with near-wall gradients in show significant changes in inclination and tilting angles of quasistreamwise vortices. These structural changes are responsible for the observed modulation of the Reynolds stress generation mechanism and the inter-component energy transfer in flows with strong near-wall gradients.
Fluids at supercritical pressure undergo a continuous phase from a liquid to a gas state if the fluid is heated above the critical pressure. During this phase transition the thermophysical properties of the fluid vary significantly within a narrow temperature range across the pseudo-critical temperature T pc Tpc
(pseudo-critical temperature is defined as the temperature at which the specific heat at constant pressure (c p cp
) attains its peak value). Figure 1 shows the variation of thermophysical properties of CO 2 2
at a thermodynamic supercritical pressure P 0 P0
= 80 bar (P critical =73.773bar Pcritical=73.773bar
) as a function of temperature (Int J Thermophys 24, 1–39 (2003)) [1]. These characteristics make supercritical fluids appealing in many industrial applications, such as: desorption, drying and cleaning in extraction processes; pharmaceutical industry; in power cycles as working fluids (Renew Sustain Energy Rev 14, 3059–3067 (2010)) [2] , (Nucl Technol 154, 283–301 (2006)) [3] and biodiesel production ( Fuel 80, 225–231 (2001)) [4].
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Fluids at supercritical pressure undergo a continuous phase from a liquid to a gas state if the fluid is heated above the critical pressure. During this phase transition the thermophysical properties of the fluid vary significantly within a narrow temperature range across the pseudo-critical temperature T pc Tpc
(pseudo-critical temperature is defined as the temperature at which the specific heat at constant pressure (c p cp
) attains its peak value). Figure 1 shows the variation of thermophysical properties of CO 2 2
at a thermodynamic supercritical pressure P 0 P0
= 80 bar (P critical =73.773bar Pcritical=73.773bar
) as a function of temperature (Int J Thermophys 24, 1–39 (2003)) [1]. These characteristics make supercritical fluids appealing in many industrial applications, such as: desorption, drying and cleaning in extraction processes; pharmaceutical industry; in power cycles as working fluids (Renew Sustain Energy Rev 14, 3059–3067 (2010)) [2] , (Nucl Technol 154, 283–301 (2006)) [3] and biodiesel production ( Fuel 80, 225–231 (2001)) [4].