Two-dimensional Inviscid Flow Research Articles

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

Registration-based nonlinear model reduction of parametrized aerodynamics problems with applications to transonic Euler and RANS flows

Exact planetary waves and jet streams

We investigate exact nonlinear waves on surfaces locally approximating the rotating sphere for two-dimensional inviscid incompressible flow. Our first system corresponds to a $\beta$ -plane approximation at the equator, and the second to a $\gamma$ approximation, with the latter describing flow near the poles. We find exact wave solutions in the Lagrangian reference frame that cannot be written down in closed form in the Eulerian reference frame. The wave particle trajectories, contours of potential vorticity and Lagrangian mean velocity take relatively simple forms. The waves possess a non-trivial Lagrangian mean flow that depends on the amplitude of the waves and on a particle label that characterizes values of constant potential vorticity. The mean flow arises due to potential vorticity conservation on fluid particles. Solutions over the entire space are generated by assuming that the flow far from the origin is zonal and there is a region of uniform potential vorticity between this zonal flow and the waves. In the $\gamma$ approximation, a class of waves is found that, based on analogous solutions on the plane, we call Ptolemaic vortex waves. The mean flow of some of these waves, which we can describe in highly nonlinear scenarios due to the exact nature of the solutions, resembles polar jet streams. Several illustrative solutions are used as initial conditions in the fully spherical rotating Navier–Stokes equations, where integration is performed via the numerical scheme presented in Salmon & Pizzo (Atmosphere, vol. 14, issue 4, 2023, 747). The potential vorticity contours found from these numerical experiments vary between stable permanent progressive form and fully turbulent flows generated by wave breaking.

Momentum, energy and vorticity balances in deep-water surface gravity waves

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

Numerical modelling of aerodynamic response to gusts and gust effect mitigation

Numerical modelling of aerodynamic response to gusts and gust effect mitigation

Accelerated construction of projection-based reduced-order models via incremental approaches

We present an accelerated greedy strategy for training of projection-based reduced-order models for parametric steady and unsteady partial differential equations. Our approach exploits hierarchical approximate proper orthogonal decomposition to speed up the construction of the empirical test space for least-square Petrov–Galerkin formulations, a progressive construction of the empirical quadrature rule based on a warm start of the non-negative least-square algorithm, and a two-fidelity sampling strategy to reduce the number of expensive greedy iterations. We illustrate the performance of our method for two test cases: a two-dimensional compressible inviscid flow past a LS89 blade at moderate Mach number, and a three-dimensional nonlinear mechanics problem to predict the long-time structural response of the standard section of a nuclear containment building under external loading.

Self-Similar-Structure–Based All-Speed Genuinely Two-Dimensional Riemann Solver

In this study, the low-speed asymptotic analysis of the Multidimensional, Self-Similar, Strongly Interacting, Consistent with the Second-Order Moment (MuSIC2) scheme in the low-speed limit is conducted. Based on such analysis, a controlling function is employed to control the numerical dissipation in the momentum equations at low speeds, and the All-Speed MuSIC2 (AMuSIC2) scheme is proposed in curvilinear coordinates. Systematic numerical tests are illustrated. The one-dimensional cases show that the AMuSIC2 scheme is capable of accurately capturing one-dimensional shocks, expansion waves, and contact discontinuities. The two-dimensional odd–even decoupling case indicates that the AMuSIC2 scheme is robust against the unphysical shock anomaly in capturing strong shocks. The spherical blast wave case indicates that the AMuSIC2 scheme improves the traditional one-dimensional Riemann solvers’ mesh imprinting phenomenon as the MuSIC2 scheme. The two-dimensional inviscid flow over the NACA0012 airfoil case and the low-speed Gresho vortex case suggest that the AMuSIC2 scheme improves the MuSCI2 scheme’s accuracy at low speeds remarkably. The turbulent flow over the flat plate case and the turbulent flow past the NACA4412 airfoil case also suggest that the AMuSIC2 scheme has a much higher level of accuracy at low speeds than its counterpart.

Multiscaled inviscid Taylor-Green vortex flow for examining energy conservation error in incompressible flows

This study investigates kinetic energy conservation errors in incompressible flows using multiplexed inviscid Taylor-Green vortices. Fourth-order precision Runge-Kutta methods, specifically five- and six-step methods, were employed and an inviscid two-dimensional periodic flow field was used as a test case. This method allows the energy conservation error to be suppressed to a very low level. Recent previous studies have dealt with turbulence generation using a multi-scale turbulence grid. In this study, the Taylor-Green vortex of the analytical solution is multiplexed to generate a flow field with a number of wavenumbers. Visualisation results of the initial field and the energy distribution over time were obtained. It was observed that the errors in the time evolution of the fluid energy were extremely small, and that the errors actually increased with time. However, the influence of the time step range was limited and the analytical and numerical solutions were in general agreement. As a result, it was confirmed that the inviscid Taylor-Green vortex is effective in examining the energy conservation error for incompressible flows.

Energy conservation uncertainly due to the use of an implicit time integration method clarified by a multiplexed TGV inviscid flow

This study attempts to elucidate the conservation errors induced by the implicit time integration method within a multiscale flow. Using the multiplexed Taylor analytical solution as the initial field, three distinct multiscale turbulence fields were formulated, each characterised by different wavenumber attributes. A two-dimensional inviscid flow field, replicated in a computational domain with a side of 2π and subject to periodic boundary conditions, was used to illuminate kinetic energy conservation errors. The implicit time integration method - specifically the second-order accurate Crank-Nicolson method - was compared with explicit methods, such as the second-order accurate Adams-Bashforth and third-order accurate Runge-Kutta methods. The study also examines the differences between the analytical Taylor solution and a random flow field, particularly in their satisfaction of the Navier-Stokes equations and wavenumber composition, while highlighting the need for preliminary analysis for random flow fields prior to validation calculations.

Registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity

Registration-based model reduction of parameterized PDEs with spatio-parameter adaptivity

Numerical simulation of bending and length-varying flapping wing using discrete vortex method

The paper aims to perform numerical simulations of flapping flight using Discrete Vortex Method (DVM) scheme. The scheme is suitable for moderately low Reynolds number [Formula: see text] and computationally less expensive as the flow domain doesn’t need to be discretized at each time step. Flapping of a one-dimensional (1D) flexible filament in a two-dimensional (2D) inviscid flow is simulated. The effect of bending by varying the wing shape along the spanwise length on aerodynamic performance was studied. It is observed that compared to rigid wings, bending wings are found to be better in generating lift. The effect of various bending wing configurations ([Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text]) and different methods of imposing the bending ([Formula: see text], [Formula: see text] and [Formula: see text]) is studied. It was demonstrated that applying wing bending only during the downstroke phase ([Formula: see text]) is more effective than imposing bending throughout the flapping cycle ([Formula: see text]). Moreover, an effort is made to replicate the bending configuration observed in manta ray fish ([Formula: see text]) to investigate its impact on flow domain characteristics, lift, and thrust forces. Furthermore, the inclusion of a winglet is found to significantly enhance lift generation. In addition to the study of bending effects on aerodynamic performance, the study also seeks to emulate a unique aspect of bat flight kinematics, specifically the dynamic variation in wing span length during flapping. In a comparative analysis of two span length variation strategies, it is discerned that exclusively varying span length during the upstroke phase is the optimal approach for achieving increased lift generation. The study highlights the crucial role of wing bending and span length modulation in achieving elevated lift forces while simultaneously reducing drag. These findings are seen as holding significant promise for the design and optimization of Micro Air Vehicles (MAVs) utilizing flapping-based lift generation mechanisms, contributing substantially to the identification of optimal parameters for enhancing MAVs’ aerodynamic performance and operational efficiency.

A conformal mapping approach to modelling two-dimensional stratified flow

Herein we describe a new approach to modelling inviscid two-dimensional stratified flows in a general domain. The approach makes use of a conformal map of the domain to a rectangle. In this transformed domain, the equations of motion are largely unaltered, and in particular Laplace's equation remains unchanged. This enables one to construct exact solutions to Laplace's equation and thereby enforce all boundary conditions. An example is provided for two-dimensional flow under the Boussinesq approximation, though the approach is much more general (albeit restricted to two-dimensions). This example is motivated by flow under a weir in a tidal estuary. Here, we discuss how to use a dynamically-evolving conformal map to model changes in the water height on either side of the weir, though the example presented keeps these heights fixed due to limitations in the computational speed to generate the conformal map. The numerical approach makes use of contour advection, wherein material buoyancy contours are advected conservatively by the local fluid velocity, while a dual contour-grid representation is used for the vorticity in order to account for vorticity generation from horizontal buoyancy gradients. This generation is accurately estimated by using the buoyancy contours directly, rather than a gridded version of the buoyancy field. The result is a highly-accurate, efficient numerical method with extremely low levels of numerical damping.

SYMMETRIES OF THE LUNDGREN–MONIN–NOVIKOV EQUATION FOR PROBABILITY OF THE VORTICITY FIELD DISTRIBUTION

A.M. Polyakov suggested the programme to expand the symmetries admitted by hydrodynamic models to the conformal invariance of statistics in the inverse cascade where the conformal group is infinite-dimensional. In the present work, the group of transformations G of the \(n\)-point probability density function fn (PDF) is presented for the infinite chain of Lundgren–Monin–Novikov equations (the statistical form of the Euler equations) for vorticity fields of the two-dimensional inviscid flow. The problem is written in the Lagrangian setting. The main result is that the group G transforms conformally the zero-vorticity characteristics and invariantly a family of the fn-equations for PDF along these lines. The equations are not invariant along other characteristics. Moreover, the action of G conserves the class of PDF. The results obtained can be used for studying the invariance of statistical properties of the optical turbulence.

Unsteady aerodynamic theory for membrane wings

We study analytically the dynamic response of membrane aerofoils subject to arbitrary, small-amplitude chord motions and transverse gusts in a two-dimensional inviscid incompressible flow. The theoretical model assumes linear deformations of an extensible membrane under constant tension, which are coupled aeroelastically to external aerodynamic loads using unsteady thin aerofoil theory. The structural and aerodynamic membrane responses are investigated for harmonic heave oscillations, an instantaneous change in angle of attack, sinusoidal transverse gusts and a sharp-edged gust. The unsteady lift responses for these scenarios produce aeroelastic extensions to the Theodorsen, Wagner, Sears and Küssner functions, respectively, for a membrane aerofoil. These extensions incorporate for the first time membrane fluid–structure interaction into the expressions for the unsteady lift response of a flexible aerofoil. The indicial responses to step changes in the angle of attack or gust profile are characterised by a slower lift response in short times relative to the classical rigid-plate response, while achieving a significantly higher asymptotic lift at long times due to aeroelastic camber. The unsteady lift for harmonic gusts or heaving motions follows closely the rigid plate lift responses at low reduced frequencies but with a reduced lift amplitude and greater phase lag. However, as the reduced frequency approaches the resonance of the fluid-loaded membrane, the lift response amplitude increases abruptly and is followed by a sharp decrease. This behaviour reveals a frequency region, controlled by the membrane tension coefficient, for which membrane aerofoils could possess substantial aerodynamic benefits over rigid aerofoils in unsteady flow conditions.

Energy conservation in the limit of filtered solutions for the 2D Euler equations

We consider energy conservation in a two-dimensional incompressible and inviscid flow through weak solutions of the filtered-Euler equations, which describe a regularized Euler flow based on a spatial filtering. We show that the energy dissipation rate for the filtered weak solution with vorticity in L p , p > 3/2 converges to zero in the limit of the filter parameter. Although the energy defined in the whole space is not finite in general, we formally extract a time-dependent part, which is well-defined for filtered solutions, from the energy and define the energy dissipation rate as its time-derivative. Moreover, the limit of the filtered weak solution is a weak solution of the Euler equations and it satisfies a local energy balance in the sense of distributions. For the case of p = 3/2, we find the same result as p > 3/2 by assuming Onsager’s critical condition for the family of the filtered solutions.

Numerical Prediction of The Gas Flows in a Reciprocating Engine Cylinder during Intake Stoke (1st Report: Inviscid Flow).(Dept.M)

This paper presents a method for predicting the in-cylinder gas flows during the intake stroke of four-stroke engines. A simplified engine model with a deep-bowl piston configuration, and a central intake valve was used. An inviscid two-dimensional axisymmetric flow was investigated. The model used a finite difference procedure with an expanding/contracting grid in axisymmetric representation. The major source of troubles for the convergence of the solution was the unsteady turbulent jet flow issuing from the intake valve which required very small time intervals for the solution compared with that during the compression stroke where there is no jet flow exists. Results show a complicated flow field during the intake stroke in engine like-geometries.

On propagation of linear waves for model of equatorial wave–current interactions

Of current mathematical and physical interest are equatorial waves, in the setting of an azimuthal two-dimensional inviscid flow in a two-layer fluid with a flat bed and a free surface. The Hamiltonian formulation for nonlinear governing equations is adequate for structure-preserving perturbations, at the linear and at the nonlinear level have been derived. Since the ocean energy is concentrated in the long-wave propagation modes, it is of interest to pursue an in-depth analysis of long waves. In the present paper we will consider the process in the absence of vorticity.

Flow stability in a wide vaneless diffuser

This work is concerned with the theoretical aspects of flow stability in a two dimensional vaneless diffuser. Specifically, the appearance of self-excited oscillations, also referred to as rotating stall, is investigated considering a two-dimensional inviscid flow in an annulus. We consider a linear perturbation method, taking as basic flow the steady potential velocity field whose radial and tangential components are inversely proportional to the radial coordinate. We show that such flow may become unstable to small two-dimensional perturbations provided that the ratio between the inlet tangential velocity and the radial one is sufficiently large and a certain amount of vorticity is injected in the flow field. Such an instability is purely kinematical, i.e. it does not involve any boundary layer effects, contrary to the classical hypothesis which ascribes the instability to a peculiar boundary layers interaction.

Large-amplitude membrane flutter in inviscid flow

We study the large-amplitude flutter of membranes (of zero bending rigidity) with vortex sheet wakes in two-dimensional inviscid fluid flows. We apply small initial deflections and track their exponential decay or growth and subsequent large-amplitude dynamics in the space of three dimensionless parameters: membrane pretension, mass density and stretching modulus. With both ends fixed, all the membranes converge to steady deflected shapes with single humps that are nearly fore-aft symmetric, except when the deformations are unrealistically large. With leading edges fixed and trailing edges free to move in the transverse direction, the membranes flutter periodically at intermediate values of mass density. As mass density increases, the motions are increasingly aperiodic, and the amplitudes increase and spatial and temporal frequencies decrease. As mass density decreases from the periodic regime, the amplitudes decrease and spatial and temporal frequencies increase until the motions become difficult to resolve numerically. With both edges free to move in the transverse direction, the membranes flutter similarly to the fixed–free case, but also translate vertically with steady, periodic or aperiodic trajectories, and with non-zero slopes that lead to small angles of attack with respect to the oncoming flow.

Arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method for computing two-dimensional compressible inviscid flows over moving bodies.

The objective of this study is to develop and apply an arbitrary Lagrangian-Eulerian unstructured finite-volume lattice-Boltzmann method (ALE-FVLBM) for solving two-dimensional compressible inviscid flows around moving bodies. The two-dimensional compressible form of the LB equation is considered and the resulting LB equation is formulated in the ALE framework on an unstructured body-fitted mesh to correctly model the body shape and properly incorporate the mesh movement due to the body motion. The spatial discretization of the resulting system of equations is performed by a second-order cell-centered finite-volume method on arbitrary quadrilateral meshes and an implicit dual-time stepping method is utilized for the time integration. To stabilize the numerical solution, appropriate numerical dissipation terms are added to the formulation. At first, the shock tube problem is computed to examine the accuracy of the solution obtained by applying the proposed FVLBM for this unsteady test case which includes shock, expansion wave, and contact discontinuity in the flow domain. Then, the stationary isentropic vortex is simulated on both the stationary and moving meshes to assess the implementation of the geometric conservation law in enhancing the solution accuracy of the ALE-FVLBM. The compressible inviscid flow in the transonic regime is then computed around the stationary NACA0012 airfoil in order to further study the sensitivity of the solution method to the user defined parameters. Now, the transonic inviscid flow is simulated over the pitching or plunging NACA0012 airfoil to investigate the accuracy and capability of the proposed solution method (ALE-FVLBM) for the computation of the compressible flows over moving bodies. Finally, the pitching or plunging NACA0012 airfoil near the ground in the transonic inviscid flow is simulated as a practical and challenging problem to study the ground effect on the aerodynamic characteristics of the airfoil. It is indicated that the solution methodology proposed based on the finite-volume LBM formulated in the arbitrary Lagrangian-Eulerian framework (ALE-FVLBM) is capable of accurately computing the compressible inviscid flows around the moving bodies with and without the ground effect.

Shock interactions in two-dimensional steady flows of Bethe–Zel’dovich–Thompson fluids

The morphology of nodes generated by the interaction of discontinuities in steady two-dimensional inviscid flows is examined. The fluids considered are Bethe–Zel’dovich–Thompson (BZT) fluids, which feature negative values of the fundamental derivative of gas dynamics in the vapour phase. The operating conditions correspond to the non-classical gas-dynamic regime where expansion shocks, compression fans and composite waves are admissible in addition to the classical compression shocks and expansion fans. Interactions caused by the crossing, overtaking and splitting of compression/expansion shocks, along with the refraction of these through a contact discontinuity, are analysed here. The well-established method of wave curves is applied to non-classical wave curves, revealing a variety of interaction patterns that are simply not admissible in classical gas dynamics. It is shown that shock waves can be reflected, transmitted and refracted as Prandtl–Meyer fans or composite waves. Based on numerical evidence, the splitting (and consequently the Mach reflection) of an expansion shock seems to be disallowed. Theoretical considerations on the admissibility of such configurations are also provided. The present analysis is relevant to applications potentially involving supersonic flows of BZT fluids, e.g. organic Rankine cycle power systems, and can also be used in front-tracking algorithms for general equations of state.