Non-isentropic Flow Research Articles

Entropy and Adjoint Methods

Aerodynamic drag can be partially approximated by the entropy flux across fluid domain boundaries with a formula due to Oswatitsch. In this paper, we build the adjoint solution that corresponds to this representation of the drag and investigate its relation to the entropy variables, which are linked to the integrated residual of the entropy transport equation. For inviscid isentropic flows, the resulting adjoint variables are identical to the entropy variables, an observation originally due to Fidkowski and Roe, while for non-isentropic flows there is a significant difference that is explicitly demonstrated with analytic solutions in the shocked quasi-1D case. Both approaches are also investigated for viscous and inviscid flows in two and three dimensions, where the adjoint equations and boundary conditions are derived. The application of both approaches to mesh adaptation is investigated, with especial emphasis on inviscid flows with shocks.

Optimal density lower bound on nonisentropic gas dynamics

In this paper, we prove a time dependent lower bound on density in the optimal order O(1/(1+t)) for the general smooth nonisentropic flow of compressible Euler equations.

Low Mach limit to one‐dimensional nonisentropic planar compressible magnetohydrodynamic equations

This paper is concerned with a one‐dimensional nonisentropic compressible planar magnetohydrodynamic flow with general initial data, whose behaviors at far fields x→±∞ are different. The low Mach limit for the system is rigorously justified. The limit relies on the uniform estimates including weighted time derivatives and an extended convergence lemma.

Numerical and experimental investigation on the effect of retrograde vaporization on fines migration and drift in porous oil reservoir: roles of phase change heat transfer and saturation

Retrograde vaporization effects on oil production are nearly unprecedented to reservoir engineering community, and its relation to formation damage should be explored. For this purpose, this paper elucidates the importance and role of this phenomenon and its phase change heat transfer (PCHT) on fines migration and subsequent, permeability damage in porous rocks bearing oil and gas. Initially, a fine particle energy conversion equation was successfully acquired by combining fine particle mass balance and general energy equations. Moreover, the computational fluid dynamic model (CFD) was adopted for performing numerical modeling. A 2D CFD model using FEA-Comsol 5.0 version was used to simulate the retrograde vaporization of reservoir fluids. Pore walls are designed as non-adiabatic, and therefore, a modified Dittus-Boelter mass transfer model is provided for a fine particle detachment under PCHT. Hence, from the simulation results it was observed that there is a high degree of heat release during reservoir fluid phase change that is from oil to gas for decreasing pressure and increasing saturation time. This heat transfer from the oil and gas influxes contributes in the expulsion and migration of in situ fines in porous media. Also, an increasing rate of enthalpy was achieved that produces a non-isentropic flow, which is required to mobilize the fines in porous medium, and a satisfactory phase transition simulation outputs were obtained and presented as well. Altogether, these factors play a significant role in the fine particle eviction from the pore chamber, thereby plugging in the pore throat and consequently, decreasing the well productivity during transient flow.

Modelling of Vane and Rotor Blade Rows in Simulations of Gas Turbine Performance

Abstract A method of modelling of nozzle and rotor blade rows of gas turbine dedicated to simulations of gas turbine performance is proposed. The method is applicable especially in early design stage when many of geometric parameters are yet subject to change. The method is based on analytical formulas derived from considerations of flow theory and from cascade experiments. It involves determination of parameters of gas flow on the mean radius of blade rows. The blade row gas exit angle, determined in turbine design point is a basis for determination of details of blade contour behind the throat position. Throat area is then fixed based on required maximum mass flow in critical conditions. Blade leading edge radius is determined based on flow inlet angle to the blade row in the design point. The accuracy of analytical formulas applied for definition of blade contour details for assumed gas exit angle was verified by comparing the results of analytical formulas with CFD simulations for an airfoil cascade. Losses of enthalpy due to non-isentropic gas flow are evaluated using the analytical model of Craig and Cox, based on cascade experiments. Effects of blade cooling flows on losses of total pressure of the gas are determined based on analytical formulas applicable to film cooling with cooling streams blowing from discrete point along blade surface, including leading and trailing edges. The losses of total pressure due to film cooling of blades are incorporated into the Craig and Cox model as additional factor modifying gas flow velocities.

Plane contours with maximum thrust in a non-isentropic flow

Plane tasks of the contour and maximum thrust nozzle modelling in the supersonic flow with shock waves are formulated and solved analytically and numerically. The first of the considered problems relates to the optimal profiling of the tail part of the body. This part consists of some corner point, followed by straight section and following him, unknown in advance curved section on which the shock interaction of the flow with the wall occurs. To improve the characteristics of the “shock” contours the exact solution of the corresponding optimization problem in the class of straight segments is considered. The second problem relates to the construction of a plane supersonic nozzle of maximum thrust in the class of contours similar to those considered in the first problem. The isentropic flow is modeled by the flow from a supersonic source. This approach allows to find the initial shape of the optimal “impact nozzle” for the subsequent improvement of its shape using a numerical method of calculation. Parametric calculations of the flow in the supersonic part of the nozzle consisting of conical and parabolic sections and its thrust are carried out which is compared with the thrust of the equivalent conical nozzle.

Study on correlation between geometrical correction and aerodynamic performance of microscale supersonic wind tunnel

AbstractIn this paper, flow visualization of the oblique shock waves of a single‐crystal silicon microscale supersonic wind tunnel is described. The microscale supersonic wind tunnel has a convergent–divergent section and a throat area of 1000 μm × 300 μm, and designed for realizing supersonic flow of Mach 2 in microscale domain. The schlieren system with integrated optical microscope was used for visualization of the oblique shock waves generated in the microscale domain. To investigate the effect of nonisentropic flow state on the inclination angle to the shock, a numerical integration of the modified Shapiro's differential equation was applied.

Global solutions to physical vacuum problem of non-isentropic viscous gaseous stars and nonlinear asymptotic stability of stationary solutions

This paper is concerned with spherically symmetric motions of non-isentropic viscous gaseous stars with self-gravitation. When the stationary entropy S‾(x) is spherically symmetric and satisfies a suitable smallness condition, the existence and properties of the stationary solutions are obtained for 65<γ<2 with weaker constraints upon S‾(x) compared with the one in [26], where γ is the adiabatic exponent. The global existence of strong solutions capturing the physical vacuum singularity that the sound speed is C12-Hölder continuous across the vacuum boundary to a simplified system for non-isentropic viscous flow with self-gravitation and the nonlinear asymptotic stability of the stationary solution are proved when 43<γ<2 with the detailed convergence rates, motivated by the results and analysis of the nonlinear asymptotic stability of Lane–Emden solutions for isentropic flows in [29,30].

Existence of Subsonic Flow in Divergent Section Adjacent to Throat of a Convergent-Divergent Nozzle for Actual Flow

In this study, a mathematical analysis by considering the effect of an actual index of expansion clearly shows a persistent of the existence of subsonic flow after the throat to a down stream in the region of divergent part to produce a supersonic velocity at the exit of a convergent-divergent nozzle. The length of the divergent part where subsonic velocity found is dependent upon the magnitude of the nozzle efficiency and the actual index of expansion. The change in velocity from subsonic, sonic and supersonic occurs only in the divergent part while the corresponding frictionless behavior has the classical features (subsonic in the convergent, sonic at the throat, and supersonic in the divergent). This is mainly due to thermodynamic processes which result a change in enthalpy due to friction and a gain in entropy. The reference conditions are newly derived for an actual frictional flow condition. This design aspect differs in a physical manner corresponding to that from an isentropic flow. An actual nozzle shape for convergent-divergent nozzles is also investigated under a non-isentropic flow condition.

A unified description of oblique waves in ideal and non-ideal steady supersonic flows around compressive and rarefactive corners

According to classical gas dynamic theory, if a steady supersonic parallel flow encounters a sudden change in the wall slope, two very different phenomena may occur. If the flow expands around a sharp corner, the well-known isentropic Prandtl–Meyer fan is observed. Conversely, a shock wave occurs if the flow is compressed: for wedge angles smaller than the detachment value, which depends on the uniform upstream state, an oblique shock originates at the corner; at larger deviation angles, a detached shock is formed. A unified description of these flows is presented here to extend the validity of the common $$\beta $$ – $$\vartheta $$ (shock angle–deflection angle) diagram for shocked non-isentropic flows into the realm of isentropic expansions. The new graph allows for a straightforward identification of the wave angles for self-similar flow fields around compressive and rarefactive corners. Besides, it clarifies the relation between shock waves and rarefaction fans in the neighbourhood of the $${\vartheta =0}$$ axis, where shock waves are weak enough to be fairly well approximated by isentropic compressions. At $${\vartheta =0}$$ , indeed, shock and rarefaction curves are demonstrated to be first order continuous. This result is interpreted in view of the bisector rule for oblique shock waves. Exemplary diagrams are reported for both ideal-gas flows, dilute-gas flows and non-ideal flows of dense vapours in the close proximity of the liquid–vapour saturation curve and critical point. The application of the new diagram is illustrated for the textbook case of the supersonic flow past a diamond-shaped airfoil.

Numerical study of two-dimensional cylindrical underwater explosion by a modified method of characteristics

A modified method of characteristics (mMOC) is introduced for solution of 2D axisymmetric supersonic flow, as well as its application to simulation of a cylindrical underwater explosion, which is a typical non-homentropic flow problem. The modifications include a family of compatibility equations which contains entropy term in streamline direction, an improved backward scheme with second-order accuracy, a modified forward scheme that uses adaptive technique, which in total make this mMOC be appropriate for a branch fluid dynamics with isentropic or non-isentropic flows. The numerical results show good agreement between the mMOC and experiments. It is established that this mMOC yields shock location, shock peak pressure, and gas-water interface location with an acceptable accuracy. Compared with an explicit dynamics program AUTODYN, which needs to select an appropriate artificial viscosity to handle shock discontinuity properly, the mMOC typically requires far less computer time with the same accuracy in the calculation of the underwater shock wave. Besides, the formation of a secondary shock is well revealed in the pressure nephogram, which suggests the mMOC is able to capture small variation in flow field. In addition, this mMOC can clearly display the domain of dependence in characteristic net, which makes it possible to quantitatively describe the connection between near field shock and detonation products expansion.

One-Dimension Nonisentropic Model for the Flow of Aluminized Explosive Products

A new analytic model of aluminized explosive products based on the method of characteristics for planar isentropic flow is proposed herein. The contribution of Al oxidation in the explosion products is investigated analytically. The flow behind the detonation front cannot be treated as isentropic due to the Al oxidation in the products. To solve the nonisentropic flow field of aluminized explosives products analytically, the assumption of local isentropic process is proposed. Based on this assumption, the flow field behind the detonation front of aluminized explosive is a function of only the reacted aluminum mass fraction in each time range. The metal plate test was conducted with the metal plate driven by RDX/Al/wax (76/20/4) and RDX/LiF/wax (76/20/4). The reacted aluminum mass can be obtained indirectly from the experiment results. The reacted aluminum mass was then applied to the analytic model, and the velocity of metal plate driven by RDX/Al/wax (76/20/4) and RDX/LiF/wax (76/20/4) was calculated. The final velocity of the metal plate driven by RDX/Al/wax was 7.8% higher than that driven by RDX/LiF/wax.

Existence of weak solutions to steady compressible magnetohydrodynamic equations

The three-dimensional equations of steady compressible magnetohydrodynamic non-isentropic flows are considered. A boundary value problem for the velocity and the temperature is studied in a bounded domain with arbitrarily large data. The existence of weak solutions to these equations is established under the assumption that the pressure P(ρ,θ)=c1ργ+c2ρθ for γ>73.

Exact solutions to non-classical steady nozzle flows of Bethe–Zel’dovich–Thompson fluids

Steady nozzle flows of Bethe–Zel’dovich–Thompson fluids – substances exhibiting non-classical gasdynamic behaviour in a finite vapour-phase thermodynamic region in close proximity to the liquid–vapour saturation curve – are examined. Non-classical flow features include rarefaction shock waves, shock waves with either upstream or downstream sonic states and split shocks. Exact solutions for a mono-component single-phase fluid expanding from a reservoir into a stationary atmosphere through a conventional converging–diverging nozzle are determined within the quasi-one-dimensional inviscid flow approximation. The novel analytical approach makes it possible to elucidate the connection between the adiabatic, possibly non-isentropic flow field and the underlying local isentropic-flow features, including the possible qualitative alterations in passing through shock waves. Contrary to previous predictions based on isentropic-flow inspection, shock disintegration is found to occur also from reservoir states corresponding to a single sonic point. The global layout of the flow configurations produced by a monotonic decrease in the ambient pressure, namely the functioning regime, is examined for reservoir conditions resulting in single-phase flows. Accordingly, a classification of steady nozzle flows into 10 different functioning regimes is proposed. Flow conditions determining the transition between the different classes of flow are investigated and each functioning regime is associated with the corresponding thermodynamic region of reservoir states.

The path decomposition technique for systems of hyperbolic conservation laws

We are concerned with the problem of the global (in time) existence of weak solutions to hyperbolic systems of conservation laws, in one spatial dimension. First, we provide a survey of the different facets of a technique that has been used in several papers in the last years: the path decomposition. Then, we report on two very recent results that have been achieved by means of suitable applications of this technique. The first one concerns a system of three equations arising in the dynamic modeling of phase transitions, the second one is the famous Euler system for nonisentropic fluid flow. In both cases, the results concern classes of initial data with possibly large total variation.

Subsonic non-isentropic Euler flows with large vorticity in axisymmetric nozzles

In this paper, we investigate the effects of vorticity for smooth subsonic non-isentropic Euler flows in infinitely long axisymmetric nozzles. It is a sequel to B. Duan and Z. Luo (2013) [13] on non-isentropic Euler system. In [13], the authors obtained the unique existence of a smooth subsonic solution with restriction on the smallness of Bernoulli's function in the upstream. The smallness assumption is sufficient to prevent the appearance of singularity, however, it leads that the vorticity is also small. The purpose of this paper is to remove such restriction on the smallness of the vorticity.

Three dimensional non-isentropic subsonic Euler flows in rectangular nozzles

In this paper, we study the existence and uniqueness of three dimensional non-isentropic subsonic Euler flows in rectangular nozzles. This work is an extension of Chen and Xie’s work on isentropic subsonic Euler flows. If, besides small normal component of vorticity, Bernoulli’s function and entropy function with small variations are given on the entrance, the existence and uniqueness of non-isentropic subsonic Euler flows are established.

Weak stability of transonic contact discontinuities in three-dimensional steady non-isentropic compressible Euler flows

In this paper, we study the stability of contact discontinuities that separate a C1 supersonic flow from a static gas, governed by the three-dimensional steady non-isentropic compressible Euler equations. The linear stability problem of this transonic contact discontinuity is formulated as a one-phase free boundary problem for a hyperbolic system with the boundary being characteristics. By calculating the Kreiss–Lopatinskii determinant for this boundary value problem, we conclude that this transonic contact discontinuity is always stable, but only in a weak sense because the Kreiss–Lopatinskii condition fails exactly at the poles of the symbols associated with the linearized hyperbolic operators. Both of the planar and nonplanar contact discontinuities are studied. We establish the energy estimates of solutions to the linearized problem at a contact discontinuity, by constructing the Kreiss symmetrizers microlocally away from the poles of the symbols, and studying the equations directly at each pole. The nonplanar case is studied by using the calculus of para-differential operators. The failure of the uniform Kreiss–Lopatinskii condition leads to a loss of derivatives of solutions in estimates.

Stability of Boundary Layers for the Nonisentropic Compressible Circularly Symmetric 2D Flow

In this paper, we study the asymptotic behavior of circularly symmetric solutions to the initial boundary value problem of the nonisentropic compressible Navier--Stokes equations in a two-dimensional exterior domain with impermeable boundary conditions when the viscosities and the heat conduction coefficient tend to zero at the same rate. By multiscale analysis, we obtain that away from the boundary the nonisentropic compressible viscous flow can be approximated by the corresponding inviscid flow, and near the boundary there are boundary layers for the angular velocity, density, and temperature in the leading order expansions of solutions, while the radial velocity and pressure do not have boundary layers in the leading order. The boundary layers of velocity and temperature are described by a nonlinear parabolic system. We prove the stability of boundary layers and rigorously justify the asymptotic behavior of solutions in the $L^{\infty}$ space for the small viscosities and heat-conduction limit in the L...

Intricacy of the Transit Manifold Concept Paid-off by Computational Accuracy

Despite its intricacy the numerical method applied within the TRANSIT code proved successful in describing discontinuous, non-isentropic flows in rocket engines and solar-gravitational towers for green energy. A number of 0-D approaches are known to render some results in demonstrating the feasibility of the solar tower concept, or in unsteady simulation of transient phases in rocket engines. Computational efficiency is demonstrated by CFD simulation of the starting transients in ADDA solid rocket engines and in the SEATTLER solar mirror tower. The code is exclusively directed to unsteady flow simulations in slender channels. The wave front model scheme covers the dual behavior of fully non-isentropic flow with mass addition and mixing in the thrust chamber or blunt heat addition in a heater and fully isentropic through the exhaust nozzle or gravity draught in a tall tower. Along the tower of the solar-gravity draught power plants small perturbation discontinuous flows are covered. Code robustness is demonstrated during runs on the PC.