Wall Mode Instability Research Articles

Wall-mode instability in plane shear flow of viscoelastic fluid over a deformable solid.

The linear stability analysis of a plane Couette flow of an Oldroyd-B viscoelastic fluid past a flexible solid medium is carried out to investigate the role of polymer addition in the stability behavior. The system consists of a viscoelastic fluid layer of thickness R, density ρ, viscosity η, relaxation time λ, and retardation time βλ flowing past a linear elastic solid medium of thickness HR, density ρ, and shear modulus G. The emphasis is on the high-Reynolds-number wall-mode instability, which has recently been shown in experiments to destabilize the laminar flow of Newtonian fluids in soft-walled tubes and channels at a significantly lower Reynolds number than that for flows in rigid conduits. For Newtonian fluids, the linear stability studies have shown that the wall modes become unstable when flow Reynolds number exceeds a certain critical value Re(c) which scales as Σ(3/4), where Reynolds number Re=ρVR/η,V is the top-plate velocity, and dimensionless parameter Σ=ρGR(2)/η(2) characterizes the fluid-solid system. For high-Reynolds-number flow, the addition of polymer tends to decrease the critical Reynolds number in comparison to that for the Newtonian fluid, indicating a destabilizing role for fluid viscoelasticity. Numerical calculations show that the critical Reynolds number could be decreased by up to a factor of 10 by the addition of small amount of polymer. The critical Reynolds number follows the same scaling Re(c)∼Σ(3/4) as the wall modes for a Newtonian fluid for very high Reynolds number. However, for moderate Reynolds number, there exists a narrow region in β-H parametric space, corresponding to very dilute polymer solution (0.9≲β<1) and thin solids (H≲1.1), in which the addition of polymer tends to increase the critical Reynolds number in comparison to the Newtonian fluid. Thus, Reynolds number and polymer properties can be tailored to either increase or decrease the critical Reynolds number for unstable modes, thus providing an additional degree of control over the laminar-turbulent transition.

Progress of the Keda Torus eXperiment Project in China: design and mission

The Keda Torus eXperiment (KTX) is a medium-sized reversed field pinch (RFP) device under construction at the University of Science and Technology of China. The KTX has a major radius of 1.4 m and a minor radius of 0.4 m with an Ohmic discharge current up to 1 MA. The expected electron density and temperature are, respectively, 2 × 1019 m−3 and 800 eV. A combination of a stainless steel vacuum chamber and a thin copper shell (with a penetration time of 20 ms) surrounding the plasma provides an opportunity for studying resistive wall mode instabilities. The unique double-C design of the KTX vacuum vessel allows access to the interior of the KTX for easy first-wall modifications and investigations of power and particle handling, a largely unexplored territory in RFP research leading to demonstration of the fusion potential of the RFP concept. An active feedback mode control system is designed and will be implemented in the second phase of the KTX program. The recent progress of this program will be presented, including the design of the vacuum vessel, magnet systems and power supplies.

Effects of multiple modes interaction on the resistive wall mode instability

The effects of multiple modes interaction on the resistive wall mode (RWM) are studied in a slab geometry with and without plasma flow. The modes interaction can have a large effect on both the linear growth rate and the nonlinear saturation level of the RWM. We found that modes interaction can suppress the linear growth rate for the most unstable mode. The plasma flow can also help to control the growth of the RWM. The RWM can be stabilized completely by a plasma flow when considering the modes interaction. The effect of modes interaction on the RWM is stronger for the mode rational surface in the vacuum than that in the plasma. The modes interaction results in a substantially lowered saturation level for the most unstable RWM.

Nonlinear instability in simulations of Large Plasma Device turbulence

Several simulations of turbulence in the Large Plasma Device (LAPD) [Gekelman et al., Rev. Sci. Instrum. 62, 2875 (1991)] are energetically analyzed and compared with each other and with the experiment. The simulations use the same model, but different axial boundary conditions. They employ either periodic, zero-value, zero-derivative, or sheath axial boundaries. The linear stability physics is different between the scenarios because the various boundary conditions allow the drift wave instability to access different axial structures, and the sheath boundary simulation contains a conducting wall mode instability which is just as unstable as the drift waves. Nevertheless, the turbulence in all the simulations is relatively similar because it is primarily driven by a robust nonlinear instability that is the same for all cases. The nonlinear instability preferentially drives k∥=0 potential energy fluctuations, which then three-wave couple to k∥≠0 potential energy fluctuations in order to access the adiabatic response to transfer their energy to kinetic energy fluctuations. The turbulence self-organizes to drive this nonlinear instability, which destroys the linear eigenmode structures, making the linear instabilities ineffective.

Stability Theory and Experiments in Wall-turbulence

Drag reduction is one of the most important areas of research. The main concept behind drag reduction is turbulence control. This could possibly be achieved in near wall turbulent flow by suppressing wall-mode instabilities (i.e., by interfering with the seeding mechanism of turbulence production). Wall-mode instabilities can be studied by applying hydrodynamic stability theory to wall bounded turbulent flows. It has been shown in Sen & Veeravalli, 2000 (Sadhana, 25, 423-437) that an anisotropic eddy viscosity model is crucial in capturing the wall mode instabilities. The main results of this theory are discussed and we present detailed experimental investigations carried out with the objective of verifying the Sen & Veeravalli theory.

Investigation of multiple roots of the resistive wall mode dispersion relation, including kinetic effects

The resistive wall mode instability in tokamak plasmas has a complex frequency which can be determined by a dispersion relation that is cubic, in general, leading to three distinct roots. A simplified model of the dispersion relation, including kinetic effects, is presented and used to explore the behavior of these roots. By changing the plasma rotation frequency, it is shown that one root has a slow mode rotation frequency (less than the inverse wall time) while the other two rotate more quickly, one leading and one lagging the plasma rotation frequency. When realistic experimental parameters from the National Spherical Torus Experiment [M. Ono et al., Nucl. Fusion 40, 557 (2000)] are used, however, only one slow rotating, near-marginal stability root is found, consistent with present experiments and more detailed calculations with the MISK code [B. Hu et al., Phys. Plasmas 12, 057301 (2005)]. Electron collisionality acts to stabilize one of the rotating roots, while ion collisionality can stabilize the other. In devices with low rotation and low collisionality, these two rotating roots may manifest themselves, but they are likely to remain stable.

Stabilization of the resistive wall mode instability by trapped energetic particles

A theoretical model for investigating the effect of the trapped energetic particles (EPs) on the resistive wall mode (RWM) instability is proposed. The results demonstrate that the trapped EPs have a dramatic stabilizing effect on the RWM because of resonant interaction between the mode and the magnetic precession drift motion of the trapped EPs. The results also show that the effect of the trapped EPs depends on the wall position. In addition, the stabilizing effect becomes stronger when the plasma rotation is taken into account. For sufficiently fast plasma rotation, the trapped EPs can lead to the complete stabilization of the RWM. Furthermore, the trapped EPs can induce a finite real frequency of the RWM in the absence of plasma rotation.

Impact of plasma poloidal rotation on resistive wall mode instability in toroidally rotating plasmas

Stability of resistive wall mode (RWM) is investigated in a cylindrical plasma and an axisymmetric toroidal plasma by taking into account not only toroidal rotation but also poloidal rotation. Since the Doppler shifted frequency is responsible for the RWM stability, the modification of this Doppler shifted frequency by poloidal rotation affects the rotation effect on RWM. When a poloidal rotation frequency is not so large, the effect of poloidal rotation on the RWM stability can be approximately treated with the modified toroidal rotation frequency. In a toroidal plasma, this modified frequency is determined by subtracting a toroidal component of the rotation parallel to the magnetic field from the toroidal rotation frequency. The poloidal rotation that counteracts the effect of the Doppler shift strongly reduces the stabilizing effect of toroidal rotation, but by changing the rotational direction, the poloidal rotation enhances this stabilizing effect. This trend is confirmed in not only a cylindrical plasma but also a toroidal plasma. This result indicates that poloidal rotation produces the dependence of the critical toroidal rotation frequency for stabilizing RWM on the rotational direction of toroidal rotation in the same magnetic configuration.

The linear model and experimentally observed resonant field amplification in tokamaks and reversed field pinches

A review is given of the experimentally observed effects related to the resonant field amplification (RFA) and the Resistive Wall Mode (RWM) instability in tokamaks and reversed field pinches (RFPs). This includes the feedback rotation of RWM in RFX-mod RFP, dependence of the RWM growth rate on the plasma-wall separation observed in JT-60U, appearance of the slowly growing RWM precursors in JT-60U and similar phenomena in other devices. The experimental results are compared with theoretical predictions based on the model comprising the Maxwell equations, Ohm’s law for the conducting wall, the boundary conditions and assumption of linear plasma response to the external magnetic perturbations. The model describes the plasma reaction to the error field as essentially depending on two factors: the plasma proximity to the RWM stability threshold and the natural rotation frequency of the plasma mode. The linear response means that these characteristics are determined by the plasma equilibrium parameters only. It is shown that the mentioned effects in different devices under different conditions can be described on a common basis with only assumption that the plasma behaves as a linear system. To extend the range of the model validation, some predictions are derived with proposals for experimental studies of the RFA dynamics.

The role of kinetic effects, including plasma rotation and energetic particles, in resistive wall mode stability

The resistive wall mode (RWM) instability in high-beta tokamaks is stabilized by energy dissipation mechanisms that depend on plasma rotation and kinetic effects. Kinetic modification of ideal stability calculated with the “MISK” code [B. Hu et al., Phys. Plasmas 12, 057301 (2005)] is outlined. For an advanced scenario ITER [R. Aymar et al., Nucl. Fusion 41, 1301 (2001)] plasma, the present calculation finds that alpha particles are required for RWM stability at presently expected levels of plasma rotation. Kinetic stabilization theory is tested in an experiment in the National Spherical Torus Experiment (NSTX) [M. Ono et al., Nucl. Fusion 40, 557 (2000)] that produced marginally stable plasmas with various energetic particle contents. Plasmas with the highest and lowest energetic particle content agree with calculations predicting that increased energetic particle pressure is stabilizing but does not alter the nonmonotonic dependence of stability on plasma rotation due to thermal particle resonances. Presently, the full MISK model, including thermal particles and an isotropic slowing-down distribution function for energetic particles, overpredicts stability in NSTX experiments. Minor alteration of either effect in the theory may yield agreement; several possibilities are discussed.

Resistive Wall Mode Instability at Intermediate Plasma Rotation

Experimental observation of resistive wall mode (RWM) instability in the National Spherical Torus Experiment (NSTX) at plasma rotation levels intermediate to the ion precession drift and ion bounce frequencies suggests that low critical rotation threshold models are insufficient. Kinetic modifications to the ideal stability criterion yield a more complex relationship between plasma rotation and RWM stability. Good agreement is found between an experimental RWM instability at intermediate plasma rotation and the RWM marginal point calculated with kinetic effects included, by the MISK code. By self-similarly scaling the experimental plasma rotation profile and the collisionality in the calculation, resonances of the mode with the precession drift and bounce frequencies are explored. Experimentally, RWMs go unstable when the plasma rotation is between the stabilizing precession drift and bounce resonances.

Model-Based Robust Control of Resistive Wall Modes via μ Synthesis

In this work, μ synthesis is employed to stabilize a model of the resistive wall mode (RWM) instability in the DIII-D tokamak. The General Atomics/FAR-TECH DIII-D RWM model, which replaces the spatial perturbation of the plasma with an equivalent perturbation of surface current on a spatially fixed plasma boundary, is used to derive a linear state-space representation of the mode dynamics. The spatial and current perturbations are equivalent in the sense that they both produce the same magnetic field perturbation at surrounding conductors. The key term in the model characterizing the magnitude of the instability is the time-varying uncertain parameter cpp, which is related to the RWM growth rate γ. Taking advantage of the structure of the state matrices, the model is reformulated into a robust control framework, with the growth rate of the RWM modeled as an uncertain parameter. A robust controller that stabilizes the system for a range of practical growth rates is proposed. The controller is tested through simulations, demonstrating significant performance increase over the classical proportional-derivative controller, extending the RWM growth rate range for which the system is stable and satisfies predefined performance constraints, and increasing the level of tolerable measurement noise. The simulation study shows that the proposed model-based DK controllers can successfully stabilize the mode when the growth rate varies over time during the discharge because of changes in the operating conditions such as pressure and rotation. In terms of robust stability, this method eliminates the need for growth-rate online identification and controller scheduling.Selected Full Papers from15th WORKSHOP ON

Weakly nonlinear stability analysis of a flow past a neo-Hookean solid at arbitrary Reynolds numbers

The finite amplitude stability of a plane Couette flow over a deformable solid medium is analyzed with emphasis on the class of high Reynolds number (Re) modes, referred to as the wall modes, for which the viscous stresses are confined to a thin layer adjacent to the fluid-solid interface with thickness O(Re−1/3) times the channel width in the limit Re⪢1. Here, the Reynolds number is defined in terms of the top plate velocity V and the channel width R. Previous linear stability analyses have shown that the wall modes are unstable for Newtonian flow past a linear viscoelastic solid. In the present study, the analysis is extended to examine the weakly nonlinear stability of these unstable wall modes in order to determine the nature of bifurcation of the transition point to finite amplitude states. To account for the finite strain deformations, the flexible solid medium is described by a neo-Hookean elastic model which is a generalization of the commonly used linear constitutive model. The linear stability analysis provides the critical shear rate Γc and the critical wavenumber in the axial direction αc, where the dimensionless shear rate is defined as Γ=ρV2/G, where ρ is the fluid density and G is the shear modulus of the elastic solid. The critical parameter Γc for the neo-Hookean solid is found to be close to Γc for the linear elastic solid analyzed in the previous studies. The first Landau constant s(1), which is the finite amplitude correction to the linear growth rate, is evaluated in the weakly nonlinear stability analysis using both the numerical technique and the high Re asymptotic analysis. The real part of the Landau constant, sr(1), is negative for the wall mode instability in the limit Re⪢1 for a wide range of dimensionless solid thickness H, indicating that there is a supercritical bifurcation of the wall mode instability. The amplitude of the supercritically bifurcated equilibrium state is derived in the vicinity of the critical point. The equilibrium amplitude, in the form A1e2/(Γ−Γc), is found to scale as Re−1/3 in the limit Re⪢1 and is proportional to H2.3 for H⪢1 in the same limit.

Robust Control of Resistive Wall Modes in Tokamak Plasmas using µ-synthesis

In this work, µ-synthesis is employed to stabilize a model of the resistive wall mode (RWM) instability in the DIII-D tokamak. The GA/Far-Tech DIII-D RWM model is used to derive a linear state space representation of the mode dynamics. The key term in the model characterizing the magnitude of the instability is the time-varying uncertain parameter c pp , which is related to the RWM growth rate γ. Taking advantage of the structure of the state matrices, the model is reformulated into a robust control framework, with the growth rate of the RWM modeled as an uncertain parameter. A robust controller that stabilizes the system for a range of practical growth rates is proposed and tested through simulations.

Adaptive Optimal Stochastic State Feedback Control of Resistive Wall Modes in Tokamaks

An adaptive optimal stochastic state feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The extended least square method with exponential forgetting factor and covariance resetting is used to identify the time-varying stochastic system model. A Kalman filter is used to estimate the system states. The estimated system states are passed on to an optimal state feedback controller to construct control inputs. The Kalman filter and the optimal state feedback controller are periodically redesigned online based on the identified system model. This adaptive controller can stabilize the time dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly four times the inverse of the growth rate for the time-invariant model used.

Adaptive stochastic output feedback control of resistive wall modes in tokamaks

An adaptive optimal stochastic output feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The system dynamics is experimentally determined via the extended least square method with an exponential forgetting factor and covariance resetting. The optimal output feedback controller is redesigned online periodically based on the system identification. The output measurements and past control inputs are used to construct new control inputs. The adaptive output controller can stabilize the time dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly three times the inverse of the growth rate. The design procedure is simpler and the computation time is shorter than the state feedback method reported earlier in Sun, Sen, and Longman [Phys. Plasmas13, 012512 (2006)].

Adaptive optimal stochastic state feedback control of resistive wall modes in tokamaks

An adaptive optimal stochastic state feedback control is developed to stabilize the resistive wall mode (RWM) instability in tokamaks. The extended least-square method with exponential forgetting factor and covariance resetting is used to identify (experimentally determine) the time-varying stochastic system model. A Kalman filter is used to estimate the system states. The estimated system states are passed on to an optimal state feedback controller to construct control inputs. The Kalman filter and the optimal state feedback controller are periodically redesigned online based on the identified system model. This adaptive controller can stabilize the time-dependent RWM in a slowly evolving tokamak discharge. This is accomplished within a time delay of roughly four times the inverse of the growth rate for the time-invariant model used.

Stability of wall modes in fluid flow past a flexible surface

The stability of wall modes in fluid flow past a flexible surface is analyzed using asymptotic and numerical methods. The fluid is Newtonian, while two different models are used to represent the flexible wall. In the first model, the flexible wall is modeled as a spring-backed, plate-membrane-type wall, while in the second model the flexible wall is considered to be an incompressible viscoelastic solid of finite thickness. In the limit of high Reynolds number (Re), the vorticity of the wall modes is confined to a region of thickness O(Re−1/3) in the fluid near the wall of the channel. An asymptotic analysis is carried out in the limit of high Reynolds number for Couette flow past a flexible surface, and the results show that wall modes are always stable in this limit if the plate-membrane wall executes motion purely normal to the surface. However, the flow is shown to be unstable in the limit of high Reynolds number when the wall can deform in the tangential direction. The asymptotic results for this case are in good agreement with the numerical solution of the complete governing stability equations. It is further shown using a scaling analysis that the high Reynolds number wall mode instability is independent of the details of the base flow velocity profile within the channel, and is dependent only on the velocity gradient of the base flow at the wall. A similar asymptotic analysis for flow past a viscoelastic medium of finite thickness indicates that the wall modes are unstable in the limit of high Reynolds number, thus showing that the wall mode instability is independent of the wall model used to represent the flexible wall. The asymptotic results for this case are in excellent agreement with a previous numerical study of Srivatsan and Kumaran.

Control of the resistive wall mode in advanced tokamak plasmas on DIII-D

Resistive wall mode (RWM) instabilities are found to be a limiting factor in advancedtokamak regimes with low internal inductance. Even small amplitude modes can affect therotation profile and the performance of these ELMing H mode discharges. Although completestabilization of the RWM by plasma rotation has not yet been observed, several dischargeswith increased beam momentum and power injection sustained good steady state performancefor record durations. The first investigation of active feedback control of the RWM hasshown promising results: the leakage of radial magnetic flux through the resistive wall canbe successfully controlled and the duration of the high beta phase can be prolonged. Theresults provide a comparative test of several approaches to active feedback control, andare being used to benchmark the analysis and computational models of active control.

A general theory for two- and three-dimensional wall-mode instabilities in boundary layers over isotropic and anisotropic compliant walls

An asymptotic theory is developed for two- and three-dimensional disturbances growing in a two-dimensional boundary layer over a compliant wall. The theory exploits the multideck structure of the boundary layer to derive asymptotic approximations at a high Reynolds number for the perturbation wall pressure and viscous stresses. These quantities can be regarded as driving the wall and, accordingly, the equation(s) of motion for the wall is (are) used as the characteristic equation(s) for finding the eigenvalue(s). The main assumptions are that the amplitude of the disturbance is sufficiently small for linear theory to hold, the Reynolds number is large, the disturbance wavelength is long compared with the boundary-layer thickness, and the critical and viscous wall layers are well separated. The theory was developed to study the travelling-wave flutter instability discussed by Carpenter and Garrad, i.e., the Class B instability of Benjamin and Landahl. Under certain limiting processes both the upper-branch and conventional triple-deck scalings for the Tollmien-Schlichting instability can be obtained with the present approach. Accordingly, the theory also gives a reliable qualitative guide to the effect of anisotropic wall compliance on the Tollmien-Schlichting instability.