The universal steady lift and drag theory and the physical origin of lift

Since most of the aerodynamic losses in turbomachinery are directly related to the various viscous flow phenomena, a reliable computer program for three-dimensional compressible viscous flows in cascades and ducts is an effective design tool for improving the performance of modern turbines and compressors. The recent remarkable advances in computer capabilities and solution algorithms enable the calculation of fully three-dimensional viscous flows and several such computer programs have been developed in recent years. Moore et al. (1979) and Hah (1983) proposed the computer programs for three-dimensional viscous compressible flows in ducts and cascades. Although both of them are available for compressible flow calculations, they are hardly applicable to transonic flow analysis.

In a recent paper, Liu et al. [“Lift and drag in three-dimensional steady viscous and compressible flow”, Phys. Fluids 29, 116105 (2017)] obtained a universal theory for the aerodynamic force on a body in three-dimensional steady flow, effective from incompressible all the way to supersonic regimes. In this theory, the total aerodynamic force can be determined solely with the vorticity distribution on a single wake plane locating in the steady linear far field. Despite the vital importance of this result, its validity and performance in practice has not been investigated yet. In this paper, we performed Reynolds-averaged Navier–Stokes simulations of subsonic, transonic, and supersonic flows over a three-dimensional wing. The aerodynamic forces obtained from the universal force theory are compared with those from the standard wall-stress integrals. The agreement between these two formulas confirms for the first time the validity of the theory in three-dimensional steady viscous and compressible flow. The good performance of the universal formula is mainly due to the fact that the turbulent viscosity in the wake is much larger than the molecular viscosity therein, which can reduce significantly the distance of the steady linear far field from the body. To further confirm the correctness of the theory, comparisons are made for the flow structures on the wake plane obtained from the analytical results and numerical simulations. The underlying physics relevant to the universality of the theory is explained by identifying different sources of vorticity in the wake.

This paper studies the lift and drag experienced by a body in a two-dimensional, viscous, compressible and steady flow. By a rigorous linear far-field theory and the Helmholtz decomposition of the velocity field, we prove that the classic lift formula $L=-{\it\rho}_{0}U{\it\Gamma}_{{\it\phi}}$, originally derived by Joukowski in 1906 for inviscid potential flow, and the drag formula $D={\it\rho}_{0}UQ_{{\it\psi}}$, derived for incompressible viscous flow by Filon in 1926, are universally true for the whole field of viscous compressible flow in a wide range of Mach number, from subsonic to supersonic flows. Here, ${\it\Gamma}_{{\it\phi}}$ and $Q_{{\it\psi}}$ denote the circulation of the longitudinal velocity component and the inflow of the transverse velocity component, respectively. We call this result the Joukowski–Filon theorem (J–F theorem for short). Thus, the steady lift and drag are always exactly determined by the values of ${\it\Gamma}_{{\it\phi}}$ and $Q_{{\it\psi}}$, no matter how complicated the near-field viscous flow surrounding the body might be. However, velocity potentials are not directly observable either experimentally or computationally, and hence neither are the J–F formulae. Thus, a testable version of the J–F formulae is also derived, which holds only in the linear far field. Due to their linear dependence on the vorticity, these formulae are also valid for statistically stationary flow, including time-averaged turbulent flow. Thus, a careful RANS (Reynolds-averaged Navier–Stokes) simulation is performed to examine the testable version of the J–F formulae for a typical airfoil flow with Reynolds number $Re=6.5\times 10^{6}$ and free Mach number $M\in [0.1,2.0]$. The results strongly support and enrich the J–F theorem. The computed Mach-number dependence of $L$ and $D$ and its underlying physics, as well as the physical implications of the theorem, are also addressed.

A finite element model is developed and used to simulate three-dimensional compressible fluid flow on a massively parallel computer. The algorithm is based on a Petrov-Galerkin weighting of the convective terms in the governing equations. The discretized time-dependent equations are solved explicitely using a second-order Runge-Kutta scheme. A high degree of parallelism has been achieved utilizing a MasPar MP-2 SIMD computer. An automated conversion program is used to translate the original FORTRAN 77 code into the FORTRAN 90 needed for parallelization. This conversion program and the use of compiler directives allows the maintenance of one version of the code for use on either vector or parallel machines. The performance of the algorithm is illustrated through its application to several example problems; execution times are presented for different computational platforms. 18 refs.

The development of computational gas dynamics (CFD) and computer technologies makes it possible to design and implement methods for computing unsteady three-dimensional viscous compressible flows in regions of complex geometry. Multigrid and preconditioning techniques allowing to speed up CFD calculations on unstructured meshes were discussed in this chapter. Flow solution was provided using cell-centered finite volume formulation of unsteady three-dimensional compressible Navier–Stokes equations on unstructured meshes. The CFD code uses an edge-based data structure to give the flexibility to run on meshes composed of a variety of cell types. The fluxes were calculated on the basis of flow variables at nodes at either end of an edge or an area associated with that edge (edge weight). The edge weights were precomputed and took into account the geometry of the cell. The nonlinear CFD solver works in an explicit time-marching fashion, based on a multistep Runge–Kutta stepping procedure and piecewise parabolic method (PPM). The governing equations were solved with MUSCL-type scheme for inviscid fluxes, and the central difference scheme of the second order for viscous fluxes. Convergence to a steady state was accelerated by the use of multigrid techniques, and by the application of block Jacobi preconditioning for high-speed flows, with a separate low Mach number preconditioning method for use with low-speed flows. The capabilities of the developed approaches were demonstrated through solving some benchmark problems on structured and unstructured meshes.

A truly three-dimensional (3D) gas-kinetic flux solver for simulation of incompressible and compressible viscous flows is presented in this work. By local reconstruction of continuous Boltzmann equation, the inviscid and viscous fluxes across the cell interface are evaluated simultaneously in the solver. Different from conventional gas-kinetic scheme, in the present work, the distribution function at cell interface is computed in a straightforward way. As an extension of our previous work (Sun et al., Journal of Computational Physics, 300 (2015) 492–519), the non-equilibrium distribution function is calculated by the difference of equilibrium distribution functions between the cell interface and its surrounding points. As a result, the distribution function at cell interface can be simply calculated and the formulations for computing the conservative flow variables and fluxes can be given explicitly. To validate the proposed flux solver, several incompressible and compressible viscous flows are simulated. Numerical results show that the current scheme can provide accurate numerical results for three-dimensional incompressible and compressible viscous flows.

A numerical technique for the analysis of three-dimensional compressible turbulent flows in a turbine stage is presented .To calculate the steady interaction flow fields in a nozzle and bucket simultaneously, the nozzle outlet elements and the bucket inlet elements are overlapped in the axial direction and are used for connecting boundary elements. To calculated the flows in arbitrarily shaped geometries, a control volume method combined with a body-fitted curvilinear coordinate system is used to obtain spatially discretized governing equations. In the present analysis, a two-equation model of turbulence is introduced to estimate the turbulence effect. In order to assure the effectiveness of the present method, a computation is carried out for the flow in a model turbine stage. Experimental data is also obtained by the use of a 5-hole pitot tube for the purpose of comparison with computational results. It is shown that three-dimensional flow phenomena due to the viscous effect are well predicted and the comparisons with experimental data give encouraging results for three-dimensional flow prediction in a turbine stage.

In a recent paper, Liu, Zhu, and Wu [“Lift and drag in two-dimensional steady viscous and compressible flow,” J. Fluid Mech. 784, 304–341 (2015)] present a force theory for a body in a two-dimensional, viscous, compressible, and steady flow. In this companion paper, we do the same for three-dimensional flows. Using the fundamental solution of the linearized Navier-Stokes equations, we improve the force formula for incompressible flows originally derived by Goldstein in 1931 and summarized by Milne-Thomson in 1968, both being far from complete, to its perfect final form, which is further proved to be universally true from subsonic to supersonic flows. We call this result the unified force theorem, which states that the forces are always determined by the vector circulation Γϕ of longitudinal velocity and the scalar inflow Qψ of transverse velocity. Since this theorem is not directly observable either experimentally or computationally, a testable version is also derived, which, however, holds only in the linear far field. We name this version the testable unified force formula. After that, a general principle to increase the lift-drag ratio is proposed.

Three-dimensional compressible–incompressible turbulent flow simulation using a pressure-based algorithm

The role of acoustics theory in Noise Control Research is reviewed. The classical sound theory of ∂2ρ/∂t2 − c2 ∂2ρ/∂xi2 = 0 which deals only with the wave propagation problem is compared with the unified acoustic theory derived by Lighthill, ∂2ρ/∂t2 − c2 ∂2ρ/∂xi2 = ∂Q/∂t − ∂Fi/∂xi + ∂2Tij/∂xi ∂xj, which deals with the sound propagation problems as well as the sound generation problems. With this unified theory, the problems of fan noise, axial compressor noise, jet noise, flow noise, and turbulent noise all become apparent. The unified theory clearly illustrates the limitation of the nearfield and farfield sound measurement that can be confidently performed. Examples of various types of noise sources and their radiation directivity will be discussed.

A viscous-inviscid zonal method is presented for calculating twoand three-dimensional compressible and incompressible external aerodynamic viscous flows. While the division of the flow field into inviscid and viscous zones is warranted by the physical nature of the problems, making full use of such division is the motivation of the present method. A finite element method is used to solve both the inviscid and viscous problems. The inviscid solution is computed by solving the potential flow with a density upwind finite element method. The viscous solution is obtained by solving the Reynolds-averaged Navier-Stokes (RANS) equations via a pressure correction algorithm plus a streamline upwind Petrov-Galerkin finite element method for space discretization. Numerical results are presented for twodimensional compressible and incompressible high Reynolds number turbulent flows around the NACA 0012 and RAE 2822 airfoils, and three-dimensional flows around the ONERA-M6 wing. The results obtained with the present viscous-inviscid zonal method are in good agreement with the full Navier-Stokes solutions of the present author or other researchers.

In this work, the three-dimensional model for the compressible micropolar fluid flow is considered, whereby it is assumed that the fluid is viscous, perfect, and heat conducting. The flow between two coaxial thermoinsulated cylinders, which leads to a cylindrically symmetric model with homogeneous boundary data for velocity, microrotation, and heat flux, is analyzed.The corresponding PDE system is formulated in the Lagrangian setting, and it is proven that this system has a generalized solution locally in time.

In this paper, non-reflecting boundary conditions based on the NSCBC method for three-dimensional compressible viscous flows, especially outlet boundary conditions, are examined and numerical tests for a curved duct flow are made. Using the subsonic non-reflecting outflow boundary conditions, curvature effects for compressible flow are investigated. First, numerical flows in a straight duct was tested for the stability of the solution and the efficiency of the subsonic non-reflecting outflow conditions extended to 3D problems. Numerical tests show that the non-reflecting NSCBC method provides accurate results for very low Reynolds number flow with isothermal no-slip walls. Second, the unsteady Navier-Stokes equations in cylindrical coordinates were numerically solved using the non-reflecting outflow conditions for a curved duct flow with isothermal no-slip walls. The numerical results show the consistency of the solution. Finally, analyzing the curved duct flow results, effects of the duct curvature were studied. A comparison of the numerical results between two curved ducts, the curvature is 102 times larger than smaller one, shows that the maximum intensity of secondary flow appeared at the outlet section of the more curved duct is about 102 times larger.

An unstructured hybrid grid method is discussed for its capability to compute three-dimensional compressible viscous flows of complex geometry. A hybrid of prismatic and tetrahedral grids is used to accurately resolve the wall boundary layers for high-Reynolds number viscous flows. The Navier-Stokes equations for compressible flows are solved by a finite volume, cell-vertex scheme. The LU-SGS implicit time integration method is used to reduce the computational time for very fine grids in boundary layer regions. Two kinds of one-equation turbulence models are evaluated here for their accuracy. The method is applied to computations of transonic flows around the ONERA M5 airplane and ONERA M6 wing, and supersonic shock/boundary layer interacting flows inside a scramjet inlet to validate the accuracy and efficiency of the method

Ninth International Conference on Numerical Methods in Fluid Dynamics