Laminar Solutions Research Articles

WIND-DRIVEN ACCRETION IN PROTOPLANETARY DISKS. I. SUPPRESSION OF THE MAGNETOROTATIONAL INSTABILITY AND LAUNCHING OF THE MAGNETOCENTRIFUGAL WIND

We perform local, vertically stratified shearing-box MHD simulations of protoplanetary disks (PPDs) at a fiducial radius of 1 AU that take into account the effects of both Ohmic resistivity and ambipolar diffusion (AD). The magnetic diffusion coefficients are evaluated self-consistently from a look-up table based on equilibrium chemistry. We first show that the inclusion of AD dramatically changes the conventional picture of layered accretion. Without net vertical magnetic field, the system evolves into a toroidal field dominated configuration with extremely weak turbulence in the far-UV ionization layer that is far too inefficient to drive rapid accretion. In the presence of a weak net vertical field (plasma beta~10^5 at midplane), we find that the MRI is completely suppressed, resulting in a fully laminar flow throughout the vertical extent of the disk. A strong magnetocentrifugal wind is launched that efficiently carries away disk angular momentum and easily accounts for the observed accretion rate in PPDs. Moreover, under a physical disk wind geometry, all the accretion flow proceeds through a strong current layer with thickness of ~0.3H that is offset from disk midplane with radial velocity of up to 0.4 times the sound speed. Both Ohmic resistivity and AD are essential for the suppression of the MRI and wind launching. The efficiency of wind transport increases with increasing net vertical magnetic flux and the penetration depth of the FUV ionization. Our laminar wind solution has important implications on planet formation and global evolution of PPDs.

Linear stability of flow in rectangular ducts in the vicinity of the critical aspect ratio

We have performed three-dimensional BiGlobal linear stability analyses of flow in rectangular ducts by using a spectral element method. A linear stability theory is applied to a steady laminar solution and an eigenvalue problem is generated in the matrix form. The critical Reynolds numbers are evaluated for some aspect ratios to describe the neutral curves. It is found that the neutral curves consist of two bifurcation branches. One is the first bifurcation branch where the flow loses its stability with respect to three-dimensional disturbances, while the other is the second bifurcation branch where it recovers its stability. The first and second branches connect at the critical point of As=3.19 and Re=2.47×105, where the saddle–node bifurcation occurs. We have revealed that the discontinuous divergence of the critical Reynolds number to infinity in the vicinity of the critical aspect ratio As is caused by the saddle–node bifurcation.

Existence of Wilton Ripples for Water Waves with Constant Vorticity and Capillary Effects

In this paper we study the water wave problem with capillary effects and constant vorticity when stagnation points are not excluded. When the constant vorticity is close to certain critical values we show that there exist Wilton ripples solutions of the water wave problem with two crests and two troughs per minimal period. They form smooth secondary bifurcation curves that emerge from primary bifurcation branches that contain a laminar flow solution and consist of symmetric waves of half of the period of the Wilton ripples, at some nonlaminar solution. We also prove that any Wilton ripple contains an internal critical layer provided its minimal period is sufficiently small.

An automatic unstructured grid generation method for viscous flow simulations

High aspect-ratio grids are required for accurate solution of boundary layer and wake flow. An approach for the efficient generation of isotropic and stretched viscous unstructured grids is introduced in this paper. The proposed grid generation algorithm starts with a very coarse initial grid. In far field regions, isotropic cells of excellent quality are produced using a combination of point insertion and cell subdivision techniques. Simultaneously, a directional grid refinement strategy is used to construct highly stretched triangular cells in viscous dominated regions. First, anisotropic unstructured grids are produced in the stream-wise direction. Then, cells close to the solid surface are refined to highly stretched layer of triangles suitable for boundary layer region. The accuracy of the current grid generation approach is assessed by laminar and turbulent compressible flow solutions around NACA0012, RAE2822, and NHLP multi-element airfoils. Numerical flow simulation results are compared with published data. Comparisons point to accuracy of the proposed unstructured viscous grid generation procedure.

Laminar instability of pressure-driven dynamos in multiple helical pipes

Dynamo action is considered in a network of intertwined, helical pipes of rectangular cross-section. The flow in each pipe is driven solely by a pressure gradient and it is assumed that both the velocity and magnetic fields remain helically symmetric as the system evolves. The exact laminar solution is followed into the nonlinear regime. In a previous study, it was found that two such pipes produced a satisfactory dynamo. However, purely hydrodynamic instabilities were found to lead to flow configurations which did not support even a kinematic dynamo. The dynamo therefore only functioned at magnetic Prandtl numbers greater than unity. In this paper, it is shown that a multi-pipe configuration drives the dynamo at a significantly lower Prandtl number. A different flow instability is observed, which is found not to inhibit magnetic field generation. A model of the flow instability is considered by analysing the Dean equations for a low-aspect-ratio rectangle. The resultant quasi-one-dimensional flow gives rise to two linked stability equations of Orr–Sommerfeld type. The inflection point in the cross-pipe secondary flow is found to drive the new instability, with the downpipe primary flow having little influence.

Equivalent and nonequivalent barotropic modes for rotating stratified flows

All the possible equivalent barotropic (EB) laminar solutions are firstly explored, and all the possible non-EB elliptic circulations and hyperbolic laminar modes of rotating stratified fluids are discovered in this paper. The EB circulations (including the vortex streets and hurricane like vortices) possess rich structures, because either the arbitrary solutions of arbitrary nonlinear Poisson equations can be used or an arbitrary two-dimensional stream function is revealed which may be broadly applied in atmospheric and oceanic dynamics, plasma physics, astrophysics and so on. The discovery of the non-EB modes disproves a known conjecture.

ON PERIODIC WATER WAVES WITH CORIOLIS EFFECTS AND ISOBARIC STREAMLINES

In this paper we prove that solutions of the f-plane approximation for equatorial geophysical deep water waves, which have the property that the pressure is constant along the streamlines and do not possess stagnation points, are Gerstner-type waves. Furthermore, for waves traveling over a flat bed, we prove that there are only laminar flow solutions with these properties.

Steady Laminar Axisymmetrical Nozzle Flow at Moderate Reynolds Numbers: Modeling and Experiment

Flow through an axisymmetrical parameterized contraction nozzle of limited size with area contraction ratio 21.8 and total length 6 cm is studied for moderate Reynolds numbers 300 < Re < 20,200. The transverse flow profiles at the nozzle exit are characterized by hot film anemometry for two different spatial step sizes. The flow at the exit is laminar and uniform in its core. Boundary layer characteristics at the nozzle exit are estimated from the transverse velocity profiles. Flow throughout the nozzle is modeled by implementing Thwaites laminar axisymmetrical boundary layer solutions in an iterative algorithm for which both universal functions, describing the shape factor and skin friction parameters respectively, are altered by adding a constant. The value of the constants is determined by fitting the modified universal functions to tabulated values reported in Blevins (Blevins, R., 1992, Applied Fluid Dynamics Handbook. Krieger, Malabar, FL.). The model is validated on the measured data. Adding nonzero constants to the universal functions improves the prediction of boundary layer characteristics so that the range of Reynolds numbers for which the discrepancy with experimental findings is less than 4% is extended from Re > 3000 to Re > 1000. Therefore, the studied contraction nozzle is of use for applications requiring a small nozzle with known low turbulence flow at the exit such as moderate Reynolds number free jet studies or bio fluid mechanics (respiration, speech production,…) and the flow at the exit of the nozzle can be accurately described by a simple boundary layer algorithm for Re > 1000.

Exponential stability of the solutions to the Boltzmann equation for the Benard problem

We complete the result in [2] by showing the exponential decay of the perturbation of the laminar solution below the critical Rayleigh number and of the convective solutions above the critical Rayleigh number, in the kinetic framework.

The laminar generalized Stokes layer and turbulent drag reduction

This paper considers plane channel flow modified by waves of spanwise velocity applied at the wall and travelling along the streamwise direction. Both laminar and turbulent regimes for the streamwise flow are studied. When the streamwise flow is laminar, it is unaffected by the spanwise flow induced by the waves. This flow is a thin, unsteady and streamwise-modulated boundary layer that can be expressed in terms of the Airy function of the first kind. We name it the generalized Stokes layer because it reduces to the classical oscillating Stokes layer in the limit of infinite wave speed. When the streamwise flow is turbulent, the laminar generalized Stokes layer solution describes well the space-averaged turbulent spanwise flow, provided that the phase speed of the waves is sufficiently different from the turbulent convection velocity, and that the time scale of the forcing is smaller than the life time of the near-wall turbulent structures. Under these conditions, the drag reduction is found to scale with the Stokes layer thickness, which renders the laminar solution instrumental for the analysis of the turbulent flow. A classification of the turbulent flow regimes induced by the waves is presented by comparing parameters related to the forcing conditions with the space and time scales of the turbulent flow.

Three-dimensional bifurcations in a cubic cavity due to buoyancy-driven natural convection

Rich and complex buoyancy-driven flow field due to natural convection will be studied numerically over a wide range of Rayleigh numbers in a cubic cavity by virtue of the simulated bifurcation diagram, limit cycle, power spectrum and phase portrait. When increasing the Rayleigh number, the predicted flow is found to evolve from the conductive state to the state with the onset of convection, which is featured with the steady and symmetric laminar solution, and then to the asymmetric state (pitchfork bifurcation), which will not be discussed in this paper. As the Rayleigh number was further increased, a limit cycle branching from the fixed point of the investigated dynamical system is observed. Supercritical Hopf bifurcation is confirmed to be the birth of the orbitally stable limit cycle that separates the vortex flow into an inner unstable region (moving away from the vortex coreline) and an outer stable region (moving towards the vortex coreline). As the Rayleigh number is increased still, the investigated buoyancy-driven flow became increasingly destabilized through quasi-periodic bifurcation and then through two predicted frequency-doubling bifurcations. Thanks to the power spectrum analysis, bifurcation scenario was confirmed to have an initially single harmonic frequency, which is featured with a driving amplitude. Then an additional ultraharmonic frequency showed its presence. Prior to chaos, in the five predicted arithmetically related frequencies there exists one frequency that is incommensurate to the other two fundamental frequencies. This computational study enlightens that the investigated nonlinear system, which involves frequency-doubling bifurcations, loses its stability to a quasi-periodic bifurcation featured with the formation of a subharmonic frequency. Subsequent to the formation of three frequency-doubling bifurcations and one quasi-periodic bifurcation, an infinite number of frequencies was observed in flow conditions with the continuously increasing Rayleigh numbers. Finally, the chaotic attractor was predicted to be evolved from the strange attractor in the corresponding phase portraits.

Hemodynamics of Ulcerated Plaques: Before and After

The hemodynamics and fluid mechanical forces in blood vessels have long been implicated in the deposition and growth of atherosclerotic plaque. Detailed information about the hemodynamics in vessels affected by significant plaque deposits can also provide insight into the mechanisms and likelihood of plaque weakening and rupture. In the current study, the governing equations are solved in their finite volume formulation in several patient-specific stenotic geometries. Of specific interest are the flow patterns and forces near ulcerations in the plaque. The flow patterns and forces in vessels with ulcerated plaques are compared with those in stenotic vessels without evidence of ulceration and to the hemodynamics in the same vessels as they likely appeared prior to ulceration. Hemodynamics "before" and "after" hemorrhage may suggest fluid mechanical and morphological factors of diagnostic and predictive value. Recirculation zones, vortex shedding, and secondary flows are captured by both laminar and turbulent solutions. The forces on vessel walls are shown to correlate with unstable plaque deposits. Performing before and after studies of vessels in long-term radiology studies may illuminate mechanisms of hemorrhage and other vessel remodeling.

Wall shear stress modulation in a turbulent flow subjected to imposed unsteadiness with adverse pressure gradient

The modulation characteristics of the turbulent wall shear stress measured in a plane diffuser subjected to imposed velocity oscillations are presented. The measurements reported in this paper pertain to nearly 150 different flows: imposed oscillations with three amplitudes and six frequencies, two different diverging channels and four streamwise positions. Only the most significant data are analyzed and discussed. The imposed unsteadiness affects the time-mean flow under the effect of the adverse pressure gradient (APG), in contrast to the canonical unsteady turbulent wall layers. The laminar viscous solution that adequately describes the amplitude and phase of the wall shear stress in channel flows in the high-imposed frequency regime is no longer valid in unsteady turbulent boundary layers subjected to APG. The latter modifies also the behavior of the phase shifts of the modulations in the turbulent quantities. The time lag of the wall shear stress turbulent intensity decreases as the pressure gradient increases. Some of these structural modifications are explained by the effect of the eddy viscosity that plays a key role in the vorticity diffusion process as the APG increases.

Slow Plasma Dynamo Driven by Electric Current Helicity in Non-Compact Riemann Surfaces of Negative Curvature

Boozer addressed the role of magnetic helicity in dynamos [1]. He pointed out that the magnetic helicity conservation implies that the dynamo action is more easily attainable if the electric potential varies over the surface of the dynamo. This provided motivated us to investigate dynamos in Riemannian curved surfaces [2]. Thiffeault and Boozer [3] discussed the onset of dissipation in kinematic dynamos. In this paper, when curvature is constant and negative, a simple laminar dynamo solution is obtained on the flow topology of a Poincare disk, whose Gauss curvature is K = –1. By considering a laminar plasma dynamo [4] the electric current helicity λ ≈ 2.34 m–1 for a Reynolds magnetic number of Rm ≈ 210 and a growth rate of magnetic field |γ| ≈ 0.022 are obtained. Negative constant curvature non-compact H2 manifold, has also been used in onecomponent electron 2D plasma by Fantoni and Tellez [5]. Chicone et al. (CMP (1997)) showed fast dynamos can be supported in compact H2. PACS: 47.65.Md.

Bifurcation and stability analysis of laminar flow in curved ducts

AbstractThe development of viscous flow in a curved duct under variation of the axial pressure gradient q is studied. We confine ourselves to two‐dimensional solutions of the Dean problem. Bifurcation diagrams are calculated for rectangular and elliptic cross sections of the duct. We detect a new branch of asymmetric solutions for the case of a rectangular cross section. Furthermore we compute paths of quadratic turning points and symmetry breaking bifurcation points under variation of the aspect ratio γ (γ=0.8…1.5). The computed diagrams extend the results presented by other authors. We succeed in finding two origins of the Hopf bifurcation. Making use of the Cayley transformation, we determine the stability of stationary laminar solutions in the case of a quadratic cross section. All the calculations were performed on a parallel computer with 32×32 processors. Copyright © 2009 John Wiley & Sons, Ltd.

Hypersonic Turbulent Flow Simulation of Fire II Reentry Vehicle Afterbody

Thispaperpresentsanumericalinvestigation ofthehypersonicreacting flowaroundtheFIREIIreentrycapsule. At the chosen freestream conditions, the forebody boundary layer and the separated flow on the afterbody are turbulent. The Reynolds-averaged Navier–Stokes method along with two commonly used turbulence models are usedtocomputethe flowfield.Accuratepredictionofturbulentseparated flowathypersonicconditionsischallenging due to the limitations of the underlying turbulence models. The presence of turbulent eddy viscosity in the flow simulation results in a smaller separation bubble than the laminar solution at identical conditions. Also, the two turbulence models predict different levels of eddy viscosity in the neck region. This has a dominant effect on the separation bubble size and the surface pressure. On the other hand, the eddy viscosity values in the near-wall region determine the heat transfer rate to the body. The two models predict comparable heating rates on the conical frustum, and the results match in-flight measurement well. By comparison, surface pressure predictions are appreciably higher than the data.

Evolution of flat-plate wakes in sink flow

Evolution of flat-plate wakes in sink flow has been studied both analytically and experimentally. For such wakes, a similarity solution is derived which considers simultaneous presence of both laminar and turbulent stresses inside the wake. This solution utilizes an additional Reynolds-stress term which represents the fluctuations similar to those in wall-bounded flows, accounting for the fluctuations originating from the plate boundary layer. In this solution, it is shown that the total stress, the sum of laminar and Reynolds shear stresses, becomes self-similar. To investigate the accuracy of the analytical results, the wake of a flat plate located at the centreline of a planar contraction is studied using hot-wire anemometry. Wakes of both tapered and blunt edges are considered. The length of the plates and the flow acceleration number K = 6.25 × 10−6 are chosen such that the boundary-layer profiles at the plate edge approach the self-similar laminar solution of Pohlhausen (Z. Angew. Math. Mech., vol. 1, 1921, p. 252). A short plate in which the boundary layer at the edge does not fully relaminarize is also considered. The development of the turbulent diffusivity used in the analysis is determined empirically for each experimental case. We have shown that the obtained similarity solutions, accounting also for the initial conditions in each case, generally agree well with the experimental results even in the near field. The results also show that the mean velocity of the transitional wake behind a tapered edge becomes self-similar almost immediately downstream of the edge.

One-dimensional model for microscale shock tube flow

A one-dimensional model for the numerical simulation of transport effects in small-scale, i.e., low Reynolds number, shock tubes is presented. The conservation equations have been integrated in the lateral directions and three-dimensional effects have been introduced as carefully controlled sources of mass, momentum and energy, into the axial conservation equations. The unsteady flow of gas behind the shock wave is reduced to a quasi-steady flow by choosing a coordinate system attached to the shock. The boundary layer problem is thereby reduced to a laminar solution, similar to the Blasius solution, with the exception that the wall velocity can be nonzero. The resulting one-dimensional equations are then solved numerically using a two-step Lax-Wendroff/ MacCormack scheme with flux correction transport. For validation purposes, comparisons are performed against previously published shock structure and low Reynolds number shock tube experiments; good agreement is observed. The model has been used to predict the performance of a 10µm shock tube and the result of this simulation shows the possibility of shock wave disappearance at lower pressure ratios for a micro-scale shock tube.

Heteroclinic connections in plane Couette flow

Plane Couette flow transitions to turbulence at Re ≈ 325 even though the laminar solution with a linear profile is linearly stable for all Re (Reynolds number). One starting point for understanding this subcritical transition is the existence of invariant sets in the state space of the Navier–Stokes equation, such as upper and lower branch equilibria and periodic and relative periodic solutions, that are distinct from the laminar solution. This article reports several heteroclinic connections between such objects and briefly describes a numerical method for locating heteroclinic connections. We show that the nature of streaks and streamwise rolls can change significantly along a heteroclinic connection.

Laminar and turbulent free convection in a composite enclosure

Turbulent natural convection in a two-dimensional horizontal composite square cavity, isothermally heated at the left side and cooled from the opposing surface, is numerically analyzed using the finite volume method. The composite square cavity is formed by three distinct regions, namely, clear, porous and solid region. The development of a numerical tool able to treat all these regions as one computational domain is of advantage for engineering design and analysis of passive thermal control systems. Governing equations are written in terms of primitive variables and are recast into a general form. It was found that the fluid begins to permeate the porous medium for values of Ra greater than 10 6. Nusselt number values show that for the range of Ra analyzed there is no significant variation between the laminar and turbulent model solution. When comparing the effects of Ra, k s/ k f and Da on Nu, results indicate that the solid phase properties have a greater influence in enhancing the overall heat transferred trough the cavity.