Nonlinear Three-dimensional Simulations Research Articles

Nonlinear Three-Dimensional Simulations of the Gradient Drift and Secondary Kelvin–Helmholtz Instabilities in Ionospheric Plasma Clouds

A newly developed three-dimensional electrostatic fluid model solving continuity and current closure equations aims to study phenomena that generate ionospheric turbulence. The model is spatially discretized using a pseudo-spectral method with full Fourier basis functions and evolved in time using a four-stage, fourth-order Runge Kutta method. The 3D numerical model is used here to investigate the behavior and evolution of ionospheric plasma clouds. This problem has historically been used to study the processes governing the evolution of the irregularities in the F region of the ionosphere. It has been shown that these artificial clouds can become unstable and structure rapidly (i.e., cascade to smaller scales transverse to the ambient magnetic field). The primary mechanism which causes this structuring of ionospheric clouds is the E×B, or the gradient drift instability (GDI). The persistence and scale sizes of the resulting structures cannot be fully explained by a two-dimensional model. Therefore, we suggest here that the inclusion of three-dimensional effects is key to a successful interpretation of mid-latitude irregularities, as well as a prerequisite for a credible simulation of these processes. We investigate the results of 2D and 3D nonlinear simulations of the GDI and secondary Kelvin–Helmholtz instability (KHI) in plasma clouds for three different regimes: highly collisional (≈200 km), collisional (≈300 km), and inertial (≈450 km). The inclusion of inertial effects permits the growth of the secondary KHI. For the three different regimes, the overall evolution of structuring of plasma cloud occurs on longer timescales in 3D simulations. The inclusion of three-dimensional effects, in particular, the ambipolar potential in the current closure equation, introduces an azimuthal “twist“ about the axis of the cloud (i.e., the magnetic field B). This azimuthal “twist” is observed in the purely collisional regime, and it causes the perturbations to have a non-flute-like character (k‖≠0). However, for the 3D inertial simulations, the cloud rapidly diffuses to a state in which the sheared azimuthal flow is substantially reduced; subsequently, the cloud becomes unstable and structures, by retaining the flute-like character of the perturbations (k‖=0).

Global modes of variable-viscosity two-phase swirling flows and their triadic resonance

The linear and nonlinear dynamics of two-phase swirling flows produced by the Grabowski–Berger profile under the influence of a viscosity stratification are investigated. We perform axisymmetric nonlinear simulations and fully three-dimensional linear global stability analysis of the flow for several swirl numbers $S$ and viscosity ratios $\tilde {\mu }$ . They are accompanied by nonlinear three-dimensional simulations and subsequent modal analysis using the bispectral mode decomposition, recently introduced by Schmidt (Nonlinear Dyn., vol. 102, issue 4, 2020, pp. 2479–2501). We find a pronounced destabilising effect of the viscosity stratification on both the onset of axisymmetric vortex breakdown and helical instability that is linked to the required shear stress continuity across the interface. Consequently, destabilisation is shifted to lower $S$ as compared with an equivalent flow with uniform viscosity. Further, the stability analysis reveals the simultaneous destabilisation of two global modes with wavenumbers $m=1$ and $m=2$ that have harmonic frequencies. The analysis of the nonlinear flow reveals a strong triadic resonance between these modes that governs the nonlinear dynamics and leads to a rapid departure from the linear dynamics. At larger swirl, the bifurcation of additional modes initiates an interaction cascade by means of triadic resonance which is elucidated by the bispectral analysis. It leads to the emergence of a variety of additional modes in the nonlinear flow. This study contributes to an improved understanding of the influence of viscosity stratification on the onset of vortex breakdown and the destabilisation of global modes. Further, it provides a clear picture of the dynamics of swirling flows with a codimension-two point and related triadic interaction of two global modes at harmonic frequencies and wavenumbers.

Centrifugal Test Replicated Numerical Model Updating for 3D Strutted Deep Excavation with the Response-Surface Method

Centrifugal tests provide an efficacious experimental process to predict the behavior of deep excavations, and numerical models are indispensable for demonstrating the test results and analyzing the engineering demand parameters. Uncertainty in material properties can cause simulations to differ from tests; therefore, updating the model becomes inevitable. This study presents a response-surface-based model updating technique for the nonlinear three-dimensional simulation of the centrifugal testing model of strutted deep excavation in sand. An overview of the fundamentals of the response-surface model is provided, including selecting uncertain parameters as input factors, creating a design order for training the model, building a second-order polynomial surface, and updating the input factors through targeted centrifugal results. The bending strains of diaphragm wall panels at multiple points along the depth are used to form the multiobjective function. Response-surface model predictions were well-matched with actual numerical responses, with less than a 0.5% difference. Parametric analyses could be conducted utilizing this updated strutted deep excavation model.

Electric Field-Induced "Tentillar" Bridging of a Droplet Twin.

Experimental investigations with high-speed imaging reveal that two unequal sized oppositely charged droplets suspended in an insulating oil can come in contact in an asymmetric manner under an electric field. The approaching poles of the droplets undergo asymmetric "cone-cone" to "cone-groove" deformations during noncoalescence under an electric field. Nonlinear three-dimensional simulations confirm the occurrence of a third "groove-groove" configuration at close proximity. A general linear stability analysis confirms the cone-cone to cone-groove transitions of the oil-water interfaces with decreasing thickness of the oil film. Experiments together with simulations confirm the bifurcation of the Taylor cone on the smaller droplet into a number of liquid "tentilla" bridges prior to contact with the bigger droplet. Simulations also predict that the length scales of the formation of such tentillar bridges from the initial Taylor cone match well with the predictions from the nonlinear bifurcation theory.

Stability of obliquely driven cavity flow

The linear stability of the incompressible flow in an infinitely extended cavity with rectangular cross-section is investigated numerically. The basic flow is driven by a lid which moves tangentially, but at yaw with respect to the edges of the cavity. As a result, the basic flow is a superposition of the classical recirculating two-dimensional lid-driven cavity flow orthogonal to a wall-bounded Couette flow. Critical Reynolds numbers computed by linear stability analysis are found to be significantly smaller than data previously reported in the literature. This finding is confirmed by independent nonlinear three-dimensional simulations. The critical Reynolds number as a function of the yaw angle is discussed for representative aspect ratios. Different instability modes are found. Independent of the yaw angle, the dominant instability mechanism is based on the local lift-up process, i.e. by the amplification of streamwise perturbations by advection of basic flow momentum perpendicular to the sheared basic flow. For small yaw angles, the instability is centrifugal, similar as for the classical lid-driven cavity. As the spanwise component of the lid velocity becomes dominant, the vortex structures of the critical mode become elongated in the direction of the bounded Couette flow with the lift-up process becoming even more important. In this case the instability is made possible by the residual recirculating part of the basic flow providing a feedback mechanism between the streamwise vortices and the streamwise velocity perturbations (streaks) they promote. In the limit when the basic flow approaches bounded Couette flow the critical Reynolds number increases very strongly.

The planar X-junction flow: stability analysis and control

AbstractThe bifurcations and control of the flow in a planar X-junction are studied via linear stability analysis and direct numerical simulations. This study reveals the instability mechanisms in a symmetric channel junction and shows how these can be stabilized or destabilized by boundary modification. We observe two bifurcations as the Reynolds number increases. They both scale with the inlet speed of the two side channels and are almost independent of the inlet speed of the main channel. Equivalently, both bifurcations appear when the recirculation zones reach a critical length. A two-dimensional stationary global mode becomes unstable first, changing the flow from a steady symmetric state to a steady asymmetric state via a pitchfork bifurcation. The core of this instability, whether defined by the structural sensitivity or by the disturbance energy production, is at the edges of the recirculation bubbles, which are located symmetrically along the walls of the downstream channel. The energy analysis shows that the first bifurcation is due to a lift-up mechanism. We develop an adjustable control strategy for the first bifurcation with distributed suction or blowing at the walls. The linearly optimal wall-normal velocity distribution is computed through a sensitivity analysis and is shown to delay the first bifurcation from $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{Re}=82.5$ to $\mathit{Re}=150$. This stabilizing effect arises because blowing at the walls weakens the wall-normal gradient of the streamwise velocity around the recirculation zone and hinders the lift-up. At the second bifurcation, a three-dimensional stationary global mode with a spanwise wavenumber of order unity becomes unstable around the asymmetric steady state. Nonlinear three-dimensional simulations at the second bifurcation display transition to a nonlinear cycle involving growth of a three-dimensional steady structure, time-periodic secondary instability and nonlinear breakdown restoring a two-dimensional flow. Finally, we show that the sensitivity to wall suction at the second bifurcation is as large as it is at the first bifurcation, providing a possible mechanism for destabilization.

Secondary Instabilities in Breaking Inertia–Gravity Waves

Abstract The three-dimensionalization of turbulence in the breaking of nearly vertically propagating inertia–gravity waves is investigated numerically using singular vector analysis applied to the Boussinesq equations linearized about three two-dimensional time-dependent basic states obtained from nonlinear simulations of breaking waves: a statically unstable wave perturbed by its leading transverse normal mode, the same wave perturbed by its leading parallel normal mode, and a statically stable wave perturbed by a leading transverse singular vector. The secondary instabilities grow through interaction with the buoyancy gradient and velocity shear in the basic state. Which growth mechanism predominates depends on the time-dependent structure of the basic state and the wavelength of the secondary perturbation. The singular vectors are compared to integrations of the linear model using random initial conditions, and the leading few singular vectors are found to be representative of the structures that emerge in the randomly initialized integrations. A main result is that the length scales of the leading secondary instabilities are an order of magnitude smaller than the wavelength of the initial wave, suggesting that the essential dynamics of the breaking might be captured by tractable nonlinear three-dimensional simulations in a relatively small triply periodic domain.

Nonlinear three-dimensional simulation of solid tumor growth

We present a new, adaptive boundary integral method to simulate solid tumor growth in 3-d. We use a reformulation of a classical model that accounts for cell-proliferation, apoptosis, cell-to-cell and cell-to-matrix adhesion. The 3-d method relies on accurate discretizations of singular surface integrals, a spatial rescaling and the use of an adaptive surface mesh. The discretized boundary integral equations are solved iteratively using GMRES and a discretized version of the Dirichlet-Neumann map, formulated in terms of a vector potential, is used to determine the normal velocity of the tumor surface. Explicit time stepping is used to update the tumor surface. We present simulations of the nonlinear evolution of growing tumors. At early times, good agreement is obtained between the results of a linear stability analysis and nonlinear simulations. At later times, linear theory is found to overpredict the growth of perturbations. Nonlinearity results in mode creation and interaction that leads to the formation of dimples and the tumor surface buckles inwards. The morphologic instability allows the tumor to increase its surface area, relative to its volume, thereby allowing the cells in the tumor bulk greater access to nutrient. This in turn allows the tumor to overcome the diffusional limitations on growth and to grow to larger sizes than would be possible if the tumor were spherical. Consequently, instability provides a means for avascular tumor invasion.

Role of current shear driven turbulence in anomalous stopping of energetic electrons

The transport and stopping of energetic electrons in a dense plasma medium is a crucial issue in the fast ignition (FI) concept of laser fusion. In this Letter nonlinear three-dimensional simulations of current shear driven electron-magnetohydrodynamics (EMHD) fluid instabilities have been presented. It is shown that these electromagnetic instabilities have a strong influence on the transport and stopping of inward directed energetic electrons. The implications of these results to the fast ignition concept of laser fusion are also discussed.

Nonlinear three-dimensional simulations of resistive wall modes

Nonlinear behavior of the resistive wall modes (RWMs) has been studied using nonlinear three-dimensional simulation code based on the reduced magnetohydrodynamic equations in a cylindrical tokamak. The vacuum is replaced by a highly resistive plasma using the pseudo-vacuum model. For the case that the KWM with (m, n) = (3, 1) mode and the internal (m, n)=(5, 2) mode are unstable, the RWM with (m, n) = (3, 1) mode enhances the growth rate of the internal (m, n) = (5, 2) mode in the nonlinear phase. The magnetic field line stochastization is obtained around the plasma cdge due to the overlapping of the magnetic islands of the (m, n) = (3, 1) mode and the (m, n) = (5, 2) mode.

Nonlinear three-dimensional simulation for self-organization and flow generation in two-fluid plasmas

A three-dimensional Hall-Magnetohydrodynamics (Hall-MHD) simulation code has been developed to study the self-organization process in two-fluid plasmas. An appreciable amount of flow is created in the direction perpendicular to the magnetic field, which is in a sharp contrast with the relaxed states in single-fluid MHD.

Dynamics of the Parker‐Jeans Instability in a Galactic Gaseous Disk

Linear analysis and nonlinear three-dimensional simulation of magnetohydrodynamics of a gas layer are carried out in local corotating coordinates, taking into account the effect of self-gravitational force. The gas is subject to the Jeans instability when there is no magnetic field, or to the Parker-Jeans instability when a magnetic field is present. We study the evolution of these instabilities in several cases which vary in factors such as the rotational speed, strength of magnetic fields, external pressure, external gravitational force, and directions of perturbations. We find that the growth rate of the instabilities and the shape of the dense blobs that aggregate in the nonlinear stage depend sensitively on these factors. Our result shows that the Jeans instability is stabilized by fast rotation while the Parker-Jeans mode may still be unstable. When the gas rotation is negligible, our nonlinear simulations show that the gas may form dense blobs or filaments that are perpendicular or parallel to the magnetic field, depending on the strength of external pressure and the direction of initial perturbations. When rotation is included, the gas forms coherent long filaments with their major axes perpendicular to the magnetic field. When we adopt typical parameters of nearby molecular clouds, the separation of these filaments is about 5 pc, consistent with the observation. This suggests that molecular clouds with a coherent filamentary structure may be attributed to the Parker-Jeans instability of a gaseous disk under the influence of rotation. This study sheds light on how interstellar/intergalactic gas aggregates to form molecular clouds and seeds of stars, and thus on the star-forming process at its very early stage.