Prandtl Number Of Order Unity Research Articles

Magnetic fields in the formation of the first stars – II. Results

ABSTRACT Beginning with cosmological initial conditions at z = 100, we simulate the effects of magnetic fields on the formation of Population III stars and compare our results with the predictions of Paper I. We use gadget-2 to follow the evolution of the system while the field is weak. We introduce a new method for treating kinematic fields by tracking the evolution of the deformation tensor. The growth rate in this stage of the simulation is lower than expected for diffuse astrophysical plasmas, which have a very low resistivity (high magnetic Prandtl number); we attribute this to the large numerical resistivity in simulations, corresponding to a magnetic Prandtl number of order unity. When the magnetic field begins to be dynamically significant in the core of the minihalo at z = 27, we map it on to a uniform grid and follow the evolution in an adaptive mesh refinement, MHD simulation in orion2. The non-linear evolution of the field in the orion2 simulation violates flux-freezing and is consistent with the theory proposed by Xu & Lazarian. The fields approach equipartition with kinetic energy at densities ∼1010–1012 cm−3. When the same calculation is carried out in orion2 with no magnetic fields, several protostars form, ranging in mass from ∼1 to 30 M⊙; with magnetic fields, only a single ∼30 M⊙ protostar forms by the end of the simulation. Magnetic fields thus suppress the formation of low-mass Pop III stars, yielding a top-heavy Pop III initial mass function and contributing to the absence of observed Pop III stars.

Global Evolution of an Accretion Disk with a Net Vertical Field: Coronal Accretion, Flux Transport, and Disk Winds

Abstract We report results from global ideal MHD simulations that study thin accretion disks (with thermal scale height H/R = 0.1 and 0.05) threaded by net vertical magnetic fields. Our computations span three orders of magnitude in radius, extend all the way to the pole, and are evolved for more than 1000 innermost orbits. We find that (1) inward accretion occurs mostly in the upper magnetically dominated regions of the disk at z ∼ R, similar to predictions from some previous analytical work and the “coronal accretion” flows found in GRMHD simulations. (2) A quasi-static global field geometry is established in which flux transport by inflows at the surface is balanced by turbulent diffusion. The resulting field is strongly pinched inwards at the surface. A steady-state advection–diffusion model, with a turbulent magnetic Prandtl number of order unity, reproduces this geometry well. (3) Weak unsteady disk winds are launched beyond the disk corona with the Alfvén radius R A /R 0 ∼ 3. Although the surface inflow is filamentary and the wind is episodic, we show that the time-averaged properties are well-described by steady-wind theory. Even with strong fields, β 0 = 103 at the midplane initially, only 5% of the angular momentum transport is driven by the wind, and the wind mass flux from the inner decade of the radius is only ∼0.4% of the mass accretion rate. (4) Within the disk, most of the accretion is driven by the Rϕ stress from the MRI and global magnetic fields. Our simulations have many applications to astrophysical accretion systems.

Partitioned second order method for magnetohydrodynamics in Elsässer variables

Magnetohydrodynamics (MHD) studies the dynamics of electrically conducting fluids, involving Navier-Stokes equations coupled with Maxwell equations via Lorentz force and Ohm's law. Monolithic methods, which solve fully coupled MHD systems, are computationally expensive. Partitioned methods, on the other hand, decouple the full system and solve subproblems in parallel, and thus reduce the computational cost. This paper is devoted to the design and analysis of a partitioned method for the MHD system in the Elsasser variables. The stability analysis shows that for magnetic Prandtl number of order unity, the method is unconditionally stable. We prove the error estimates and present computational tests that support the theory.

Heat transfer during simultaneous impact of two drops onto a hot solid substrate

A numerical method based on the Volume-of-Fluid approach has been used for simulating the simultaneous collision of two drops with a solid substrate. Heat transfer in the substrate and in the drop have been evaluated during the drop spreading and receding phases. The numerical model includes the liquid evaporation from the drop surface and especially in the neighborhood of the moving contact line. The evolution of the heat transfer rate at the liquid/substrate interface is also modelled theoretically by considering the development of the thermal boundary layers in the solid wall and in the spreading drop. The theoretical model does not take into account the effect of drop evaporation on the overall heat transport. It is shown that at high Prandtl numbers the heat flow is mainly determined by the instantaneous wetted area while for Prandtl numbers of order unity, the contribution of evaporation is significant. As a result, at high values of Prandtl number the agreement between the theoretical model and numerical prediction is significantly better than at the Prandtl number values of order unity.

Numerical simulation of a dual-source supersonic plasma jet expansion process: continuum approach

Expanding thermal plasma (ETP) is a versatile technology for thin film deposition process with directional plasma flux and high deposition rates. This process involves expansion of supersonic plasma jets through a steep pressure ratio into a chamber maintained at near vacuum. Usually the plasma jets deviate from chemical and thermal equilibrium and the continuum approach is insufficient to describe the phenomena. In the current work, the continuum approach based Navier–Stokes equations have been implemented to study and understand the jet expansion process in a typical dual-arc plasma deposition reactor. The numerical predictions have been compared against in-house experimental data obtained by thermocouple measurements. For the range of back pressures (6–200 Pa) considered, it was observed that the jet core is supersonic and transitions to a subsonic zone downstream without the formation of any Mach disc for the prevalent operating parameters. Indications of thick and smeared barrel shocks were however observed in the computed flow-thermal fields. The modelled fluid was assumed to be a perfect gas with temperature dependent specific heats, thermal conductivity and viscosity coefficients, with constant Prandtl number of order unity. The radial spreads of the jets increase with increasing pressure ratio thus leading to enhanced interactions within reduced distances downstream of the nozzle exit. The jet core Mach number also increases, but moderately, with decreasing backpressure. It is concluded that within reasonable accuracy, continuum approach based calculations are able to capture most of the important phenomena involved in compressible, high-temperature, supersonic jet expansion processes which are essential in designing chambers relevant to the mentioned processes.

Mean profiles for a passive scalar in wall-bounded flows from symmetry analysis

Based on the symmetry properties of the equations for a passive scalar in turbulent wall-bounded flows we derive the symmetry invariant mean profiles of the scalar. In a unifying framework we classify all of the coexisting profiles in terms of their invariance properties and derive analytical expressions for the different scaling laws of the mean profiles for velocity and passive scalars at Prandtl numbers of order unity. In addition to the well-known logarithmic law also algebraic and exponential profiles are found. The results are supported by extensive numerical simulations in channel and Couette flows as well as experiments on thermal boundary layers.

The magnetic geometry of magnetic-dominated thin accretion disks

The presence of an imposed external magnetic field drastically influences the structure of thin accretion disks. The magnetic field energy is assumed to be in balance with the thermal energy of the accretion flow, and a diusion approximation simulates the (vertical) energy transport. Our main result is that inside the corotation radius the resulting radial inclination i of the magnetic field lines from the rotation axis easily exceeds the critical value 30 (required to launch cold jets) even for mag- netic Prandtl numbers of order unity. The self-consistent consideration of both magnetic field and accretion flow demonstrates only a weak dependence of the inclination angle on the magnetic Prandtl number. The surface values of the toroidal magnetic fields necessary to induce considerably high values for the radial inclination are much smaller than expected. As the innermost part of the disk produces the largest B tor , the largest radial inclination can also be expected there. The idea is therefore supported that the cold jets are basically launched in the central disk area.

MHD instability in differentially-rotating cylindric flows

The possibility that the magnetic shear-flow instability (MRI, also Balbus-Hawley instability) might give rise to turbulence in a cylindrical Couette flow is investigated through numerical simulations. The study is linear and the fluid is assumed to be incompressible and dierentially rotating with the rotation law = a+b=R 2 . The model is fully global in all three spatial directions with boundaries on each side; nite diusivities are also imposed. The computations are carried out for several values of the azimuthal wavenumber m of the perturbations in order to analyze whether or not non-axisymmetric modes are preferred, which in a nonlinear extension of the study nally might lead to a dynamo-generated magnetic eld. For magnetic Prandtl number of order unity we nd that with a magnetic eld the instability is generally easier to excite than without a magnetic eld. The critical Reynolds number for Pm = 1 is of the order of 50, independent of whether or not the nonmagnetic flow is stable. We nd that i) the magnetic eld strongly reduces the number of Taylor vortices, ii) the angular momentum is transported outwards and iii) for nite cylinders a net dynamo-alpha eect results which is negative (positive) for the upper (lower) part of the cylinder. For magnetic Prandtl number smaller than unity the critical Reynolds number appears to scale with Pm 0:65 . If this was true even for very small magnetic Prandtl numbers (e.g. for 10 5 , the magnetic Prandtl number of liquid sodium) the critical Reynolds number should reach the value of 10 5 which, however, is also characteristic of the nonlinear nite-amplitude hydrodynamic Taylor-Couette turbulence { so that we have to expect the simultaneous existence of both sorts of instabilities in related experiments. Similar phenomena are also discussed for cold accretion disks with their basically small magnetic Prandtl numbers.

Magnetic field dragging and the vertical structure of thin accretion discs

ABSTRA C T The presence of an imposed vertical magnetic field may drastically influence the structure of thin accretion discs. If the field is sufficiently strong, the rotation law can depart from the Keplerian one. We consider the structure of a disc for a given eddy magnetic diffusivity but neglect details of the energy transport. The magnetic field is assumed to be in balance with the internal energy of the accretion flow. The thickness of the disc as well as the turbulent magnetic Prandtl number and the viscosity, a , are the key parameters of our model. The calculations show that the radial velocity can reach the sound speed for a magnetic disc if the thickness is comparable to that of a non-magnetic one. This leads to a strong amplification of the accretion rate for a given surface density. The inclination angle of the magnetic field lines can exceed the critical value 308 (required to launch cold jets) even for a relatively small magnetic Prandtl number of order unity. The toroidal magnetic fields induced at the disc surface are smaller than predicted in previous studies.

Heat Transfer From a Circular Cylinder at Low Reynolds Numbers

A correlation formula, Nu = W0(Re)Pr1/3 + W1(Re), that is valid in a wide range of Reynolds and Prandtl numbers has been developed based on the asymptotic expansion for Pr → ∞ for the forced heat convection from a circular cylinder. For large Prandtl numbers, the boundary layer theory for the energy equation is applied and compared with the numerical solutions of the full Navier Stokes equations for the flow field and energy equation. It is shown that the two-terms asymptotic approximation can be used to calculate the Nusselt number even for Prandtl numbers of order unity to a high degree of accuracy. The formulas for coefficients W0 and W1, are provided.

The Slip-Dominated Merged Layer Regime for Newtonian Hypersonic Flow Past a Flat Plate

The rarefied uniform hypersonic flow past the leading edge of a sharp flat plate at zero angle of attack is analyzed on the basis of a continuum model consisting of the Navier–Stokes equations and the velocity-slip and temperature-jump plate boundary conditions. The model fluid is a perfect gas having constant specific heats, a constant Prandtl number of order unity, and first and second viscosity coefficients varying as a power of the absolute temperature. For this flow, it is taken that the Newtonian parameter, $\varepsilon = ( {\gamma - 1} )/( {\gamma + 1} )$, goes to zero, and that the freestream Mach number, $M = ( \rho _\infty u_\infty ^2 /\gamma p_\infty )^{1/2} $, the stagnation temperature parameter, $\theta _S = \{ 1 + \varepsilon )/( 1 - \varepsilon )\}\varepsilon M^2 $, and the free-stream Reynolds number (based on the characteristic axial length from the leading edge), $R_L = \rho _\infty u_\infty L/\mu _\infty $, go to infinity.For the viscosity-temperature exponent, $\omega $, satisfying $1...

The Three-Dimensional Steady Radial Expansion of a Viscous Gas from a Sonic Source into a Vacuum

The three-dimensional steady radial expansion of a viscous, heat-conducting, compressible fluid from a spherical sonic source into a vacuum is analyzed using the Navier–Stokes equations as a basis. It is assumed that the model fluid is a perfect gas having constant specific heats, a constant Prandtl number of order unity, and viscosity coefficients varying as a power of the absolute temperature. Limiting forms for the flow variable solutions are studied for the Reynolds number based on the sonic source conditions, $1/\delta $, going to infinity and the Newtonian parameter, $\varepsilon $, both fixed and going to zero.For the case of the viscosity-temperature exponent, $\omega $, $1/2\leqq \omega < 1$, it is shown that the velocity as well as the pressure approach zero as the radial distance goes to infinity.The formulations of the distinct regions that span the domain extending from the sonic source to the vacuum are presented. For $\delta \to 0$, $\varepsilon $ fixed, there are three such regions; for $\...

Hypersonic Strong-Interaction Solutions for Flow Past a Very Slender Axisymmetric Body

The Navier–Stokes hypersonic strong-interaction theory is presented for the flow of a viscous, heat conducting, compressible fluid past a very slender axisymmetric body, when the ratio of the radial thickness of the body to the radial thickness of the viscous region, produced and supported by the body, goes to zero. The fluid is assumed to be a perfect gas having constant specific heats,a constant Prandtl number of order unity, and viscosity coefficients varying as a power of the absolute temperature. Solutions are studied for the free stream Mach number, the free stream Reynolds number based on the body’s axial length, and a modified hypersonic viscous interaction parameter going to infinity. It is shown that, for a viscosity-temperature exponent, $\omega $, less than one, five distinct layers span the region between the shock wave and the body. The leading approximations for the behavior of the flow in each of these five regions are analyzed, and it is found that an arbitrary body supports $3/ 4$-power viscous and shock layers.

The failure of a boundary layer model to describe certain cases of cellular convection

A model has been developed to describe the two-dimensional cellular motion of a viscous fluid with Prandtl number of order unity heated from below at high Rayleigh number. This model is easily adapted to describe such flow for infinite Prandtl number provided all boundaries are free. It has been suggested that a good approximation to the physical problem of flow between rigid horizontal boundaries will be obtained by setting the Prandtl number equal to infinity, thus ignoring the momentum convection. Similar cellular motion occurs in flow in a porous medium at high Rayleigh numbers and a similar model has been proposed to describe this motion. The basic features of the model are outlined and it is shown that the model fails to describe either the motion of an infinite Prandtl number fluid between parallel rigid boundaries or fluid in a porous medium. An analysis of the model for large finite Prandtl number and large Rayleigh number shows that as the Prandtl number is increased the velocity boundary layers on the rigid horizontal boundaries thicken, and eventually fill the cell, thus losing their boundary layer identity and causing the breakdown of the model.

Hypersonic weak-interaction solutions for flow past a very slender axisymmetric body

The Navier–Stokes hypersonic weak-interaction theory is presented for the flow of a viscous, heat-conducting, compressible fluid past a very slender axisymmetric body, when the ratio of the radius of the body to the radial thickness of the viscous region, produced and supported by the body, is much less than unity. The fluid is assumed to be a perfect gas having constant specific heats, a constant Prandtl number of order unity, and viscosity coefficients varying as a power of the absolute temperature. Solutions are studied for the free-stream Mach number, the free-stream Reynolds number based on the axial length of the body, and the reciprocal of the weak-interaction parameter much greater than unity.It is shown that, for the viscosity-temperature exponent ω less than 1, seven distinct layers span the region between the shock wave and the body, which is of arbitrary shape. The leading approximations for the behaviour of the flow in these seven layers are analyzed, and the restrictions imposed on the theory are obtained.

Finite amplitude convection cells

In this paper we consider two-dimensional steady cellular motion in a fluid heated from below at large Rayleigh number and Prandtl number of order unity. This is a boundary-layer problem and has been considered by Weinbaum (1964) for the case of rigid boundaries and circular cross-section. Here we consider cells of rectangular cross-section with three sets of velocity boundary conditions: all boundaries free, rigid horizontal boundaries and free vertical boundaries (referred to here as periodic rigid boundary conditions), and all boundaries rigid; the vertical boundaries of the cells are insulated. It is shown that the geometry of the cell cross-section is important, such steady motion being not possible in the case of free boundaries and circular cross-section; also that the dependence of the variables of the problem on the Rayleigh number is determined by the balances in the vertical boundary layers.We assume only those boundary layers necessary to satisfy the boundary conditions and obtain a Nusselt number dependence $N \sim R^{\frac{1}{3}}$ for free vertical boundaries. For the periodic rigid case, Pillow (1952) has assumed that the buoyancy torque is balanced by the shear stress on the horizontal boundaries; this is equivalent to assuming velocity boundary layers beside the vertical boundaries (rather than the vorticity boundary layers demanded by the boundary conditions) and leads to a Nusselt number dependence N ∼ R¼. If it is assumed that the flow will adjust itself to give the maximum heat flux possible the two models are found to be appropriate for different ranges of the Rayleigh number and there is good agreement with experiment.An error in the application of Rayleigh's method in this paper is noted and the correct method for carrying the boundary-layer solutions round the corners is given. Estimates of the Nusselt numbers for the various boundary conditions are obtained, and these are compared with the computed results of Fromm (1965). The relevance of the present work to the theory of turbulent convection is discussed and it is suggested that neglect of the momentum convection term, as in the mean field equations, leads to a decrease in the heat flux at very high Rayleigh numbers. A physical argument is given to derive Gill's model for convection in a vertical slot from the Batchelor model, which is appropriate in the present work.