Finite Pressure Gradient Research Articles

Scaling studies of the high mode pedestal

The structure and scaling of the H-mode (high mode) pedestal are examined for discharges in the DIII-D tokamak [Plasma Physics and Controlled Fusion Research (International Atomic Energy Agency, Vienna, 1986), p. 159]. For typical conditions, the pedestal values of the ion and electron temperatures Ti and Te are comparable. Measurements of main ion and C6+ profiles indicate that the ion pressure gradient in the barrier is 50%–100% of the electron pressure gradient for deuterium plasmas. The magnitude of the pressure gradient in the barrier often exceeds the predictions of infinite-n ballooning mode theory by a factor of 2. Moreover, via the bootstrap current, the finite pressure gradient acts to entirely remove ballooning stability limits for typical discharges. For a large dataset, the width of the pressure barrier δ is best described by the dimensionless scaling δ/R∝(βpolped)0.4 where (βpolped) is the pedestal value of poloidal beta and R is the major radius. Scalings based on the poloidal ion gyroradius or the edge density gradient do not adequately describe overall trends in the data set and the propagation of the pressure barrier observed between edge-localized modes. The width of the Ti barrier is quite variable and is not a good measure of the width of the pressure barrier.

Optimum safety factor profile for an advanced tokamak with negative central shear

The effect of the wall location on the stability of magnetohydrodynamic equilibria is considered for the case of an advanced tokamak with negative central magnetic shear (NCS). The equilibria have pressure and current profiles consistent with the proposed second stable core VH (SSC-VH) mode. In particular, the pressure probe has a finite edge pressure gradient that is consistent with high confinement as seen in experiments. The stability analysis shows that, for q0 = 3.9, all qmin in the range 1.8 ⩽ qmin ⩽ 3.3 are stable to n=1 and n=2 modes at βN = 5.0 with the DIII-D wall. However, there are two optimum minimum q (qmin) with respect to the wall distance from the plasma. For n=1 modes with q0 = 3.9, the optimum qmin is 2.1, with a secondary optimum at qmin = 2.85. The wall position for the n = 2 mode is more restrictive than that for the n = 1 mode but the optimum qmin values for both n = 1 and n = 2 are almost identical

Theory of low Mach number compressible flow in a channel

The properties of a relatively uncommon regime of fluid dynamics, low Mach number compressible flow are investigated. This regime, which is characterized by an exceptionally large channel aspect ratio L/d ∼ 106 leads to highly subsonic flows in which friction dominates inertia. Even so, because of the large aspect ratio, finite pressure, temperature, and density gradients are required, implying that compressibility effects are also important. Analytical results are presented which show, somewhat unexpectedly, that for forced channel flow, steady-state solutions exist only below a critical value of heat input. Above this value the flow reverses against the direction of the applied pressure gradient causing fluid to leave both the inlet and outlet implying that the related concepts of a steady-state friction factor and heat transfer coefficient have no validity.

Relaxed state of a toroidal fusion plasma with stationary flows

Relaxation of a plasma in the presence of zeroth-order flows is studied in a realistic toroidal geometry. The field-aligned flow allows equilibria with finite pressure gradient but homogeneous temperature distribution. Numerical calculations have been performed for the first and second ellipticity regimes of the extended Grad–Shafranov–Schlüter equation. Two distinct classes of solutions exist, depending on the value of the safety factor: one with horizontally stratified density and pressure and one with concentric surfaces of constant density. A comparison of these results is made with those of Finn and Antonsen.

Minimum-Energy State of an Axisymmetric Toroidal Plasma

A new axisymmetric relaxed equilibrium model in the form of Grad-Shafranov equations is proposed. Utilizing the constraint condition \( K_{\omega 0}=\int _{V_{\rm p}} \omega {\mib A} \cdot {\mib B} {\rm d}v\) developed by Bhattacharjee and Dewar instead of the helicity, this model includes a finite pressure gradient and satisfies the practical plasma boundary condition that current density at the plasma surface become zero J ( r p )=0. We compare this model with others in the limit of a large-aspect-ratio toroidal plasma, and this model is shown to simulate other cylindrical reversed field pinch models as well as a wider class of configurations, such as diamagnetic and skin current types.

Nonlinear analysis of flow in an elastic tube (artery): steady streaming effects

We analyse the nonlinear flow of a Newtonian fluid in an elastic tube when subjected to an oscillatory pressure gradient with motivation from the problem of blood flow in arteries. Two parameters: the unsteadiness, α = R 0 (ω/ν) ½ and the diameter variation, e = ( R max − R 0 )/ R 0 , are important in characterizing the flow problem. The diameter variation (e) is taken to be small so that the perturbation method is valid, and asymptotic solutions for two limiting cases of the steady-streaming Reynolds number, R s = (αe) 2 (either small or large), are derived. The results indicate that nonlinear convective acceleration induces finite mean pressure gradient and mean wall shear rate even when no mean flow occurs. The magnitude of this effect depends on the amplitude of the diameter variation and the flow rate waveforms and the phase angle difference between them, which can be related to the impedance (pressure/flow) phase angle. Changes in the impedance phase angle, which is indicative of the degree of wave reflection, can change the direction of the induced mean flow. It is also shown that the induced mean wall shear rate is proportional to α when α is large. In addition, it is observed that the steady flow structure in the core can be influenced by wave reflection. The streamlines in the core are always parallel to the tube wall when there is no reflection. However, with total reflection, the induced mean flow recirculates between the nodes and points of maximum amplitude in a closed streamline pattern. Implications of the steady-streaming phenomena for physiological flow applications are discussed in a concluding section.

On the Pressure Gradient Force over Steep Topography in Sigma Coordinate Ocean Models

Abstract The error in computing the pressure gradient force near steep topography using terms following (σ) coordinates is investigated in an ocean model using the family of vertical differencing schemes proposed by Arakawa and Suarez. The truncation error is estimated by substituting known buoyancy profiles into the finite difference hydrostatic and pressure gradient terms. The error due to “hydrostatic inconsistency,” which is not simply a space truncation error, is also documented. The results show that the pressure gradient error is spread throughout the water column, and it is sensitive to the vertical resolution and to the placement of the grid points relative to the vertical structure of the buoyancy field being modeled. Removing a reference state, as suggested for the atmosphere by Gary, reduces the truncation error associated with the two lowest vertical modes by a factor of 2 to 3. As an example, the error in computing the pressure gradient using a standard 10-level primitive equation model appl...

Calculation of turbulent boundary layers with transpiration using a second-moment closure.

The development of turbulent boundary layers with blowing or suction in zero or finite pressure gradients is calculated using a second-moment closure that is applicable right up to a wall. The turbulence model was previously shown to give good predictions for several types of flow, including a rotating channel flow. In the present calculation, no special changes are made to deal with transpiration, except for a wall boundary condition fro the normal velocity. Published experimental data on the friction coefficient, the mean velocity profiles, and the turbulent normal stress profiles are used to assess the performance of the model calculation. It is shown that the numerical solutions are generally in agreement with the data. This paper thus shows the wide applicability of the turbulence model.

Containment forces in stable plasma configurations

It is demonstrated that finite pressure gradients can be maintained in the lowest free energy state (‘relaxed state’) of a plasma configuration. The pressure forces are supported by the Magnus Force in a compressible adiabatic plasma if mass motions are considered. It has been demonstrated elsewhere that these forces can be of the same order of magnitude as the Lorentz forces in some laboratory plasmas.

A turbulent dynamo model for the reversed field pinches of plasma

A dynamo model for reversed field pinches (RFP’s) is proposed from the viewpoint of magnetohydrodynamic (MHD) turbulence. This model aims at investigating the sustainment phase of RFP’s, and consists of the mean magnetic induction equation and three equations for the bulk properties of MHD fluctuations, namely, the MHD turbulent energy, its dissipation rate, and the MHD helicity. In particular, the alpha dynamo effect is composed of the familiar alpha term and a new alpha term closely related to inhomogeneity of the field. This model paves the way for explaining the generation and sustainment of alpha and beta effects. Moreover, it can treat finite plasma pressure gradients and fulfill the vanishing of plasma currents at the plasma edge. This model can also be extended to the Earth’s magnetic dynamo model where the mean motion of a fluid plays an important role.

Theoretical study of onset conditions for solar eruptive processes

We apply the model of quasistatic equilibrium sequences to describe the time development of magnetic field structures in the plasma of the solar corona, and to determine onset points of a dynamical evolution. The representation of the magnetic field by Euler potentials provides a realistic modeling of the photospheric boundary conditions. We present a numerical method suited for the computation of magnetohydrodynamic equilibrium states and for analysing their stability against perturbations within ideal MHD. Pressure and magnetic footpoint displacement can be prescribed separately as boundary conditions. We consider magnetic arcade structures typical for large two-ribbon flares. Our results indicate that a finite pressure gradient seems to be essential for the existence of onset points. Furthermore, it is shown that magnetic shear destabilizes for intermediate values, but can have a stabilizing effect for a large amount of shear.

Self-consistent dynamolike activity in turbulent plasmas.

The evolution of turbulent plasmas is investigated within the framework of resistive magnetohydrodynamics. The functional form of the mean electric field is derived for fluctuations generated by tearing and resistive-interchange modes. It is shown that a bath of such local and global modes in pinches causes toroidal field reversal with finite pressure gradients in the plasma.

A Class of Airfoils Having Finite Trailing-Edge Pressure Gradients

Some new developments relevant to the design of single-element airfoils using potential flow methods are presented. In particular, the ramifications of the unbounded trailing edge pressure gradients generally present in the potential flow solution for the flow over an airfoil are examined, and the conditions necessary to obtain a class of airfoils having finite trailing edge pressure gradients developed. The incorporation of these conditions into the inverse method of Eppler for the design of low-speed airfoils is discussed, and designs generated using the modified scheme are presented for consideration. A detailed viscous analysis of one of these airfoils demonstrates a significant reduction in the strong inviscid-viscid interactions generally present near the trailing edge. These reductions offer the possibility of improved airfoil performance, as well as the possibility of improved accuracy in the methods of airfoil design and analysis.

Flow of cohesive solids in standpipes against a pressure gradient

For downflow of cohesive solids in a standpipe in the absence of pressure gradient, the flow, no-flow criterion of Jenike (1964) can be applied to determine the minimum diameter for no arching to occur. In this paper Jenike's analysis is extended to account for the presence of pressure gradient in the standpipe. An equation for estimating minimum pipe diameter for smooth flow is presented. For design we suggest that the minimum diameter in the absence of pressure gradient should be measured and our equation is to be applied to extrapolate to the real situation with a finite pressure gradient. The present work is relevant to recent usage of standpipes in oil-shale processing and coal power plants.

Two-and-a-half-dimensional magnetohydrodynamic turbulence

Incompressible magnetohydrodynamic turbulence is considered in a geometry in which the fields are all independent of the z coordinate, e.g., but have all three components. This is a limit frequently called ’’two-and-a-half-dimensional.’’ It is conjectured that simultaneous inverse cascades of magnetic helicity and mean square z-component of vector potential may exist. At high Reynolds numbers, selective decays to states of finite pressure gradients appear to be possible. Anti-dynamo theorems are difficult to prove for this geometry.

Helical Symmetric Reversed Field Configurations with Minimum Potential Energy

The Helical symmetric reversed field configurations with minimum potential energy has been studied by applying the variational principles to the cylindrical plasma of finite plasma pressure.The helical symmetric magnetic field with finite pressure gradient, which is derived from the MHD equilibrium equation, can't satisfy the condition of minimum potential energy obtained based on the Woltjer's invariance.An axisymmetric RFP which satisfies the Woltjer's invariance, can only confine the pressure gradient plasma without destroying the MHD equilibrium.

Stability aspects of plasmas penetrated by neutral gas with respect to velocity driven modes

A study of the stability properties of dense partially ionized plasmas immersed in strong magnetic fields with respect to velocity driven modes is presented. The investigation is limited to modes with characteristic frequencies larger than the electron and ion diamagnetic drift frequencies but less than the ion gyro frequency. The analysis is applicable to high density rotating plasma experiments with considerable mass motion also along the magnetic field. The mass motions perpendicular and parallel to the lines of force are associated with externally applied electric fields and finite pressure gradients respectively. The presence of a third fluid, neutral gas can give rise to strong stabilizing effects within certain parameter regions also for modes actually driven by plasma-neutral gas interaction effects.

Adsorption in non-isobaric fixed beds

The paper examines the effect of finite pressure gradients on the dispersion of a pulse of an adsorbable gaseous species as it flows through a fixed bed of adsorbing spherical particles.It is shown that, even for significant deviations from isobaric conditions within the bed, there is an asymptotic linear relationship between the first moment of the response (the mean retention time) and the reciprocal of the exit flow rate, a characteristic of the isobaric situation. However, unlike the isobaric case, extrapolation of the asymptotic linearity to infinite flow rate leads to a non-zero first moment at that point. Furthermore, extrapolation to zero first moment corresponds to a hypothetical flow rate which is dependent only on the parameters of the packed bed and the viscosity of the carrier gas and, specifically, is independent of the nature of the adsorbed species. The theory has been confirmed by a series of chromatographic adsorption experiments under non-isobaric conditions.Subsequently, the same experimental data have been employed in conjunction with the theory to obtain adsorption equilibrium constants for four normal alkanes (propane, butane, pentane, and hexane) on alumina in the temperature range 50–200°C with an uncertainty of ±3%.

Propagation of Signals of Finite Concentration in Gas Chromatography. III. Influence of the Pressure Gradient

Abstract Resolution of the equations of the quasi-ideal chromatography model at finite concentration is possible in the case of a finite pressure gradient in the column provided the elution does not change the pressure profile and the pressure variation inside the solute band is neglected. A study of the solution shows that there still exists three models of peak migration, as in the zero pressure gradient case, but that the pressure gradient enhances the isotherm effect. It is not possible, however, to account for the effect of pressure gradient by using an average pressure because there is a convolution between the effect of the pressure gradient and a nonlinear isotherm. Analysis of the double-shock propagation mode and the determination of the limiting transition conditions permit accurate measurement of the curvature of the isotherm at infinite dilution. The experimental results (variation of retention time with peak height for n-pentane in squalane) are in excellent agreement with the predictions of...

Numerical Prediction of the Pipeline Flow Characteristics of Thixotropic Liquids

Abstract A numerical technique has been developed to permit establishing the pressure gradient associated with laminar flow of thixotropic liquids through long pipelines. For this purpose the pipeline is divided into a number of radial and longitudinal increments within which rheological properties of the fluid may be considered as constant at any time. Then, provided only that the fluid flow curve is defined at every duration of shear, it is possible to predict the instantaneous pressure gradient at any cross-section along the pipeline for each desired flow rate and pipe size. The technique consists of an iterative integration of shear rate to establish the appropriate value of the wall shear stress at each location. Consistency of fluid in the increment is determined by the flow history of that increment, while the radial flow) associated with variations in velocity profile is accounted for by adjusting the width and radial position of the increment. A number of pressure profiles, computed at each of several flow rates, provide a convenient basis for pipeline design and pump selection. Introduction In recent years, considerable attention has been given to predicting pressure drop associated with the isothermal laminar flow of time-independent non-Newtonian fluids in pipes and annuli. The approach generally has been m develop analytical relationships between flow rate and pressure drop based on simple constitutive models which hopefully provide an approximate description of the rheological properties of the fluid. Analytical solutions are highly desirable since the influence of all pertinent parameters can be readily determined. Unfortunately, however, this approach is restricted to simple flow geometries and frequently leads to erroneous results due to inadequacies in the model. In certain cases a solution may be obtained through applying appropriate numerical techniques For example, a digital computer program is available for predicting the velocity profile and pressure drop encountered by any Newtonian or time-independent non-Newtonian fluid flowing under laminar conditions in a cylindrical pipe or annulus. In this paper the consistency behavior of the fluid need only be described in terms of basic rheological data. Analyzing flow systems involving fluids with time-dependent rheological characteristics is considerably more complicated since substantial changes in consistency may occur because of sustained shear action. This sensitivity to shear frequently persists for several hours. Consequently, variations in pressure drop and/or flow rate resulting from the aging process and addition of unsheared or partially sheared fluid to the system must be considered for purposes of pipeline design. This paper outlines a numerical method for predicting the transient and steady-state laminar flow behavior of a thixotropic liquid in a pipeline of arbitrary length (i.e., at a specified constant flow rate, the instantaneous pressure gradient may be determined at any time after start up and at any location along the pipeline). Several such pressure gradient profiles computed at several flow rates, may be combined to produce a complete portrait of the system response. This flow portrait provides a reasonable basis for pipeline design and for selecting a suitable pump characteristic. TIME-DEPENDENT RHEOLOGICAL BEHAVIOR The most familiar time-dependent rheological properties are those exhibited by thixotropic liquids. Many of these materials, particularly thixotropic crude oils, generally display an apparent yield stress in that a finite pressure gradient is required to initiate flow. Then, under the influence of sustained shear at a constant shear rate, the consistency systematically decreases to some final limiting value. SPEJ P. 369ˆ