General Compressible Flows Research Articles

Continuous and discontinuous compressible flows in a converging–diverging channel solved by physics-informed neural networks without exogenous data

Physics-informed neural networks (PINNs) are employed to solve the classical compressible flow problem in a converging–diverging nozzle. This problem represents a typical example described by the Euler equations, a thorough understanding of which serves as a guide for solving more general compressible flows. Given a geometry of the channel, analytical solutions for the steady states do indeed exist, and they depend on the ratio between the back pressure of the outlet and the stagnation pressure of the inlet. Moreover, in the diverging region, the solution may branch into subsonic flow, supersonic flow, or a mixture of both with a discontinuous transition where a normal shock occurs. Classical numerical schemes with shock fitting and capturing methods have been developed to solve this type of problem effectively, whereas the original PINNs are unable to predict the flows correctly. We make a first attempt to exploit the power of PINNs to solve this problem directly by adjusting the weights of different components of the loss function to acquire physical solutions and in the meantime, avoid trivial solutions. With a universal setting yet no exogenous data, we are able to solve this problem accurately; that is, for different given pressure ratios, PINNs provide different branches of solutions at both steady and unsteady states, some of which are discontinuous in nature. For an inverse problem such as unknown specific-heat ratio, it works effectively as well.

Dynamic simulation on flow characteristics of KSTAR PF magnet cryogenic network

During its nominal operation, strong current variation, more than 10 kA per second, can be imposed on the KSTAR PF (Poloidal Field) magnet system. Due to AC loss, transient massive backward flow can occur, especially at the cryogen inlets of each magnet. To protect cryogen circulator, several valves are activated to adjust or lower pressure below the circulator's operation limit. In this work, we discuss whether currently available thermo-hydraulic code, such as SUPERMAGNET, can describe this overall dynamic flow characteristics of KSTAR PF magnet cryogenic network. First, three major functions of the KSTAR PF magnet cryogenic system during plasma operation are classified and discussed. Cryogenic components mostly related with other functions, such as cool-down process, are removed from the simulation circuit to minimize computation time. SUPERMAGNET code was used with a slight modification in its cryogenic network simulation module, FLOWER. The codes for cryogenic valves are modified so that not only steady state but also general compressible flows can be described. A case study for an actual KSTAR plasma operation has been intensively carried out. It was shown that most relevant data, pressure, temperature and mass-flow, at the inlets and outlets of the magnet, are in agreement with simulation results.

Derivation of fractional-derivative models of multiphase fluid flows in porous media

This paper is devoted to deriving several fractional-order models for multiphase flows in porous media, focusing on some special cases of the two-phase flow. We derive the mass and momentum conservation laws of multiphase flow in porous media. The mass conservation-law has been developed based on the flux variation using Taylor series approximation. The fractional Taylor series's advantage is that it can represent the non-linear flux with more accuracy than the first-order linear Taylor series. The divergence term in the mass conservation equation becomes of a fractional type. The model has been developed for the general compressible flow, and the incompressible case is highlighted as a particular case. As a verification, the model can easily collapse to the traditional mass conservation equation once we select the integer-order. To complete the flow model, we present Darcy’s law (momentum conservation law in porous media) with time/space fractional memory. The modified Darcy’s law with time memory has also been considered. This version of Darcy’s law assumes that the permeability diminishes with time, which has a delay effect on the flow; therefore, the flow seems to have a time memory. The fractional Darcy’s law with space memory based on Caputo's fractional derivative is also considered to represent the nonlinear momentum flux. Then, we focus on some cases of fractional time memory of two-phase flows with countercurrent-imbibition mechanisms. Five cases are considered, namely, traditional mass equation and fractional Darcy’s law with time memory; fractional mass equation with conventional Darcy’s law; fractional mass equation and fractional Darcy’s law with space memory; fractional mass equation and fractional Darcy’s law with time memory; and traditional mass equation and fractional Darcy’s law with spatial memory.

점성 효과를 반영한 압축성 와류 유동의 선형 안정성 해석기법 개발

Physical understanding of vortex flow instability is one of the essential prerequisites for proper prediction and analysis of the behavior of vortex flow encountered in many engineering problems. In this study, a stability analysis code for axisymmetric vortex flow is developed based on the linear stability theory. A stability equation is derived from the governing equations for general compressible flows in the cylindrical coordinate system in order to take into account both the effects of compressibility and viscosity on stability results. The code is validated through analyses on widely-known model vortex problems by comparing the results with those from preceding researches. The comparison of the results confirms that variations of unstable eigenvalues with respect to vortex and disturbance parameters as well as the maximum amplification rate can be obtainable very accurately from the developed code. The effect of viscosity on the instability characteristics of supersonic vortex flow was further analyzed by carrying out the additional analysis. The results show that the analysis of the present study provide a more accurate prediction on linear amplification of disturbances compared to the inviscid analyses carried out in former studies. This also indicates that the present code can be considerably useful for parametric studies on the instability of compressible vortex flows with viscous effects.

Compressible Navier-Stokes Equations in Cylindrical Passages and General Dynamics of Surfaces—(I)-Flow Structures and (II)-Analyzing Biomembranes under Static and Dynamic Conditions

A new approach to solve the compressible Navier-Stokes equations in cylindrical co-ordinates using Geometric Algebra is proposed. This work was recently initiated by corresponding author of this current work, and in contrast due to a now complete geometrical analysis, particularly, two dimensionless parameters are now introduced whose correct definition depends on the scaling invariance of the N-S equations and the one parameter δ defines an equation in density which can be solved for in the tube, and a geometric Variational Calculus approach showing that the total energy of an existing wave vortex in the tube is made up of kinetic energy by vortex movement and internal energy produced by the friction against the wall of the tube. Density of a flowing gas or vapour varies along the length of the tube due to frictional losses along the tube implying that there is a pressure loss and a corresponding density decrease. After reducing the N-S equations to a single PDE, it is here proven that a Hunter-Saxton wave vortex exists along the wall of the tube due to a vorticity argument. The reduced problem shows finite-time blowup as the two parameters δ and α approach zero. A rearranged form for density is valid for δ approaching infinity for the case of incompressible flow proving positive for the existence of smooth solutions to the cylindrical Navier-Stokes equations. Finally we propose a CMS (Calculus of Moving Surfaces)–invariant variational calculus to analyze general dynamic surfaces of Riemannian 2-Manifolds in R 3 . Establishing fluid structures in general compressible flows and analyzing membranes in such flows for example flows with dynamic membranes immersed in fluid (vapour or gas) with vorticity as, for example, in the lungs there can prove to be a strong connection between fluid and solid mechanics.

Exact theory of material spike formation in flow separation

We develop a frame-invariant theory of material spike formation during flow separation over a no-slip boundary in two-dimensional flows with arbitrary time dependence. Based on the exact curvature evolution of near-wall material lines, our theory identifies both fixed and moving flow separation, is effective also over short time intervals, and admits a rigorous instantaneous limit. As a byproduct, we derive explicit formulae for the evolution of material line curvature and the curvature rate for general compressible flows. The material backbone that we identify acts first as the precursor and later as the centrepiece of unsteady Lagrangian flow separation. We also discover a previously undetected spiking point where the backbone of separation connects to the boundary, and derive wall-based analytical formulae for its location. Finally, our theory explains the perception of off-wall separation in unsteady flows and provides conditions under which such a perception is justified. We illustrate our results on several analytical and experimental flows.

Lagrangian Transport through Surfaces in Compressible Flows

A material-based, i.e., Lagrangian, methodology for exact integration of flux by volume-preserving flows through a surface has been developed recently in [Karrasch, SIAM J. Appl. Math., 76 (2016), pp. 1178-1190]. In the present paper, we first generalize this framework to general compressible flows, thereby solving the donating region problem in full generality. Second, we demonstrate the efficacy of this approach on a slightly idealized version of a classic two-dimensional mixing problem: transport in a cross-channel micromixer, as considered recently in [Balasuriya, SIAM J. Appl. Dyn. Syst., 16 (2017), pp. 1015-1044].

Unsteady fluid flow separation by the method of averaging

We use the method of averaging to improve recent separation criteria for two-dimensional unsteady fluid flows with no-slip boundaries. Our results apply to general compressible flows that admit a well-defined asymptotic average. Such flows include periodic and quasiperiodic flows, as well as aperiodic flows with a mean component. As an example, we predict and verify the unsteady separation location and angle in variants of an oscillating separation bubble model.

A segregated method for compressible flow computation. Part II: general divariant compressible flows

Typically, segregated methods have been used for the computation of incompressible flows whereas coupled solvers, for compressible flows. Compared to coupled solvers, segregated methods present the advantage of computational savings in RAM memory and CPU time, although at the cost of an inferior robustness. However, previously published segregated algorithms for general compressible flows are known to present pitfalls, like convergence to wrong solutions, lack of robustness in the presence of strong discontinuities, such as normal and oblique shocks, and complicated boundary condition imposition. Therefore, a segregated method for non-isothermal compressible flows is proposed that preserves the thermodynamic coupling and overcomes the criticisms of existing methods.

Consistent Boundary Conditions for Reduced Navier-Stokes (RNS) Scheme Applied to Three-Dimensional Viscous Flows

A consistent and efficient set of boundary conditions are developed for the multi-sweep space marching pressure-elliptic Reduced Navier-Stokes (RNS) scheme as applied for three-dimensional internal viscous flow problems. No-slip boundary conditions are directly imposed on the solid walls. There is no iteration procedure required in the cross plane to ensure mass conservation across each marching plane. The finite difference equations forming the coefficient matrix are ordered such that the surface normal velocity is specified on all the solid walls; unlike external flows, a pressure boundary condition in the cross plane is not required. Since continuity is directly satisfied at all points in the flow domain, the first order momentum equations can be solved directly for the pressure without the need for a Poisson pressure correction equation. The procedure developed herein can also be applied with periodic boundary conditions. The analysis is given for general compressible flows. Incompressible flow solutions are obtained, for straight and curved ducts of square cross section, to validate the procedure. The solutions of these test cases are used to demonstrate the applicability of the RNS scheme, with the improved boundary conditions, for internal flows with strong interaction as would be encountered in ducts and turbomachinery geometries.

An appraisal of ‘flat plate’ closure for the approximate solution of boundary layer problems

SummaryThe usefulness of zero pressure gradient, flat plate closure relations in providing approximate solutions for the boundary layer momentum and energy integral equations is examined. Expressions are obtained for skin friction, surface heat transfer rate and local Reynolds analogy factor under general compressible flow conditions. For laminar flows the predictions are compared with well known similarity solutions, with some exact solutions for non-similar flows and with experimental heat transfer data for high speed flow over a blunt cone. Consideration is also given to situations in which the surface temperature is a function of position. For turbulent flow situations comparisons are made with experimental data obtained from two-dimensional and axi-symmetric tests. Conditions vary from low Mach number incompressible flows through to high Mach number compressible flows with highly cooled walls. Some comparisons are also made with other prediction techniques.

Flows with heat addition

The present paper is a review of the work on heat addition with which the author has been concerned. It deals first with some general principles of heat addition, both at low Mach number in two dimensions and in general compressible flow in one dimension, with either constant or slowly varying cross section. A general two-dimensional method is then described with some examples of propulsion for a waverider aircraft. The special case of isothermal heat addition is discussed, and a general discussion is given of heat addition associated with separated flow, as in a flame-holder.

A Study of Compressible Potential and Asymptotic Viscous Flows for Comer Region

A new and direct approach for calculating the first higher order potential flow along an axial corner is presented. For the incompressible potential flow, the present approach demonstrates that the displacement effects in the corner may be visualized as the superposition of the displacement effects for the two intersecting semi-infinite plates forming the corner. The compressible subsonic potential flow is then obtained by the Prandtl-Glauert rule. Linearized airfoil theory is used to determine the potential flow for the supersonic case. The asymptotic viscous flow, to lowest order, for the corner problem has been calculated for general compressible flow. The analysis presented here recovers all the previously obtained lowest order asymptotic solutions. Cross flow velocity profiles have been given for M f between 0.001 to 4 for an adiabatic wall as well as for a prescribed temperature at the wall. The behavior of the cross flow skin-friction coefficient is shown to be quite different from that of the skin friction coefficient due to classical axial flow.