Partial Differential Equations based on their mathematical behavior can exhibit totally different solutions in the flow field. The different and unique behavior of PDEs depends on their characteristics. It is very important to study these characteristics of PDEs so that we can make an assessment of the kind of numerical method that should be applicable for solving them. Any valid numerical solution must exhibit the property of obeying the general mathematical properties of the governing equations. The governing equation can be classified as either Parabolic, Elliptical, Hyperbolic, or as Mixed PDE.
Characteristic Lines of PDE
PDE can be classified based on their characteristics. The characteristics of PDE are lines across which we could have discontinuities in the highest order derivative.
Consider the system of the quasi-linear equations as shown above. u & v are the continuous function of x & y. We can imagine that u & v represent the continuous velocity field in xy plane. At any given point in xy plane, we have a unique value of u & v, moreover, the derivatives of u & v i.e. du/dx, du/dy, dv/dx & dv/dy are finite in xy space.
However, there are special lines in xy plane across which derivatives of u & v are indeterminant and across which may be discontinuous. These special lines are Characteristics Lines. In other words, we consider any point C in xy plane, let us find out such lines or direction(if exist) in xy plane that passes through point C along which derivatives of u & v are discontinuous or indeterminant.
The existence of these Characteristics Lines is the identifiable curve in the xy plane. Therefore we should be able to calculate the equation for these curves, and especially the slope of these curves through point C. Those equations (of characteristics lines) would help in classifying the PDEs. Methods like Cramer’s Rule & Eigen Value are used to classify these partial differential equations.
Following the Cramer’s Rule, We will end with equation like:
The above equation is a quadratic equation for dy/dx. a, b & c are the function of the coefficients (a1,b1,c1,d1,a2,b2,c2,d2) shown in above the system of quasi-linear equations. Solution of the above equation in xy plane will give the slopes of lines along which derivatives of u & v are indeterminant. So the slope of the characteristics through point C can be calculated as:-
The above equation gives the direction of the characteristic lines through point C in xy plane, These lines have different nature depending on the discrimination D = b2 – 4ac.
If D > 0 – Two real characteristics exist through point C in XY plane. Such equations are known as Hyperbolic equations.
If D = 0 – Such equations are called Parabolic equations.
if D < 0 – The characteristics curves are imaginary and the equations are called Elliptical equations.
Eigen Value Method
We can classify the quasi-linear partial differential equations using Eigen Value Method.
We can define W column vector as:-
Now we can define our above PDEs in a Matrix form.
[K] * [M] are 2×2 matrix. Multiplying the equation with [K]-1
Where [N] = [K]-1[M]
Now the eigenvalue of matrix [N] determines the classification of PDE. If the eigenvalue is all real then the equations are hyperbolic, If the eigenvalue is complex then the equations are elliptical.
Consider the 2D irrotational, 2D, inviscid, steady flow of a compressible gas. If the flow field is slightly perturbed from its freestream conditions, maybe flow over a thin aerofoil at a small AOA and if the freestream Mach no. is either subsonic or supersonic then the governing continuity, momentum, and energy equation reduces to:-
We can see the classification of these equations by applying Eigen Value Method. Finally, we will end up knowing that the classification of these equations depends on Freestream Mach no. If the M<1 subsonic flow then the governing equations are classified as Elliptical Equation & if the M>1 supersonic flow then the governing equations are classified as Hyperbolic Equation.
For some Systems of equations, the eigenvalue could be complex of both real and complex values. Such equations exhibit a Mixed Elliptical-Hyperbolic behavior equation. Full Naiver Stokes Equation is one such example.
Characteristics of PDEs are very important to understand from mathematical and physical perspectives. It is very important to understand how the mathematical characteristic of PDE reflects physics. Each type of equation has different mathematical behavior and this reflects the different physical behavior of the flow field as well. Hence we need to develop the computational method that obeys the mathematical behavior of these equations and gives a valid solution of course the problem must be well-posed.
2nd Order Partial Differential Equation
Hyperbolic equations are special equations with 2 characteristic lines that define the “Domain of Dependence” and “Region of Influence”. Any disturbance at point P will only be felt in Region I which is a Region of Influence. The solution of hyperbolic Equations is set up as a time marching solution either in space if the problem is steady or in time if the problem is unsteady with known initial conditions. Steady Euler Equations are hyperbolic equations when the local Mach no. is supersonic & Unsteady Euler Equations are always hyperbolic in time no matter what the local Mach no. is.
Steady Inviscid Supersonic Flow is governed by Steady Euler Hyperbolic equations in which flow is supersonic everywhere, hence it is governed by hyperbolic equation. We need to define the initial flow field data upstream from the region of interest. Then the solution can be computed numerically by marching step by step in the flow direction. Steady supersonic flow over an aerofoil are an example of such a problem.
Unsteady Inviscid Flows governing Euler equations are always hyperbolic no matter what the local Mach number is. Such flows are hyperbolic with respect to time, meaning marching direction is always time and we need to define initial flow field data in complete space at time t = 0 seconds. 1D wave motion in a duct & 2D Unsteady flow over a flapping airfoil is an example of such a problem.
Hyperbolic equations with two characteristics across which we can have possible discontinuities are valid for supersonic flows. Disturbance propagates in all directions at a finite speed of the characteristic of the medium. At supersonic speed, disturbance propagates at a speed which is not matching with the source propagation speed, meaning the source is moving much faster than the disturbance. So disturbance dissipation is insignificant which leads to the accumulation of disturbance. Due to too much accumulation, there will be a rapid discontinuity over which these disturbances are released via shock wave formation. That leads to a jump in property of the flow and shock formation. In this case, the choice of numerical method must be able to capture these discontinuities. That’s why it is very important to know the characteristics of these equations. Since the disturbance propagates at a finite speed, it will propagate up to a certain region. Outside that region disturbance won’t be felt. That region is separated by lines (in 2d), these lines are the characteristic of hyperbolic PDE. There comes discontinuity across these lines.
Solution of Parabolic Equation deals with Initial-Boundary Value problems. With known boundary and initial condition solution in the domain is computed. Starting from the initial condition solution is marched either in time or space. Parabolic Equations lend themselves to marching type solutions, analogous to hyperbolic equations.
Steady Boundary Layer flow is an example for the parabolic equation. When the Reynolds number based on body length is large and assuming the boundary layer is thin then the Navier-Stokes Equation can be approximated as Boundary Layer Equations. These boundary layer equations are parabolic equations. In such an application we assume all viscous effects are contained in the boundary layer and flow outside the boundary layer is inviscid. High-Speed flow over an aerofoil is one such example where within the boundary layer simplified Boundary Layer equations can be solved. As shown in the above picture of Boundary Layer Flow on an aerofoil, boundary condition comes from the aerofoil surface “no-slip condition” and the outer edge of the boundary layer where “inviscid flow conditions” are defined. We need to define that initial data of the flow field may be near the leading edge of the aerofoil, such initial data can be obtained from the self-similar solution for a flat wedge surface or for a sharp cone. Having defined initial and boundary conditions, we march forward along the aerofoil direction to compute the solution in the boundary layer.
Unsteady heat conduction problem is governed by Parabolic equation i.e. parabolic in time. Assume we have fluid separated by two parallel plates, Initially, fluid and plate temperature are the same and the system is in equilibrium. Then at t = 0, the Temperature of the right plate is impulsively increased, the Temperature field of that instant is the initial condition, and the temperature of the left and right plates is the boundary condition. So it is an initial-boundary value problem. As a result, there will be transient variation in temperature distribution due to thermal condition which is governed by unsteady thermal conduction parabolic equation. At the Boundary, we impose a disturbance that’s the occurrence of discontinuity which propagates in all directions with infinite speed. Parabolic Equation with time marching problem has one discontinuity. We have an initial condition. Discontinuity is felt in the domain as time marches because disturbance is imposed at the boundary of the domain and propagates in all directions with infinite speed. Due to the significant amount of dissipation, the system will try to homogenize the disturbance in the domain that is imposed at the boundary.
The characteristics curve for the Elliptical Equation is imaginary. We don’t have a limited region of influence or domain of dependence, rather information is propagated everywhere in all directions. These are the major characteristics of the Elliptical equation. Let us consider a closed domain abcd as shown in the above picture, If we introduce a disturbance at point P, the disturbance will be felt everywhere throughout the domain. Therefore solution at point P has to be computed simultaneously with the solution of all other points in the domain abcd. Such a solution approach is totally different from the “marching approach” of parabolic and hyperbolic equations. We need to specify boundary conditions for all boundaries in the computational domain. These boundary conditions can be Dirichlet or Newmann or Mixed type. So this is a Boundary-Value Problem.
Steady Inviscid Subsonic Flow is governed by Elliptical Equation. Steady Euler Equations for Mach number less than 1 are Elliptical in Nature. In subsonic flow, disturbance propagates faster than the source and it propagates to infinity in all directions. Such is the pattern of streamline for subsonic flow over aerofoil as shown above. Notice that Streamlines in front of the aerofoil deflected upward and behind the aerofoil deflected downward. Disturbance introduced by the presence of aerofoil is felt throughout the entire flow field. The above picture is a picture consistent with the mathematical behavior of the elliptical equation.
Steady-state conduction is an example of an Elliptical equation. In the elliptical equation, disturbance propagates in all directions at infinite speed. The solution in the domain is always smooth/continuous because disturbance propagates in all directions at infinite speed. (Boundary value problem) The boundary can have discontinuities so the effect of discontinuities from the boundary dissipates very rapidly & Disturbance felt and dissipates rapidly in the entire domain.
The above table summarizes the classification of Second Order PDE: Type of Problem, Equation type, Prototype equation, Conditions, Solution Domain, and Solution smoothness associated with Elliptic, Parabolic, and Hyperbolic Equation.
In this article we have talked about various simplified, approximated forms of Navier-Stokes Equation(with their different mathematical behavior) depending on the particular flow field to be analyzed like Euler Equation, Boundary-Layer Equation, Conduction Equation etc. The full Navier-Stokes equation exhibits mixed mathematical behavior. Therefore when we talk about what type of fluid flow fields are governed by Parabolic equations, we are really talking about what types of approximated flow-field models are governed by Parabolic equations. The same philosophy is followed for Hyperbolic and Elliptical equations. The full Navier-Stokes equation has both Parabolic and Elliptical behavior. The parabolic behavior comes from the time derivatives of velocity and internal energy just like unsteady heat conduction and the elliptical behavior comes from the viscous term which provides the mechanics for feeding information upstream in the flow. The time-Marching approach is a well-posed, most numerical solution to the full, compressible Navier Stokes equations using the “time marching approach”.
Reference: Chapter-3 Computational Fluid Dynamics by John D. Anderson, Chapter-2 An introduction to CFD by Versteeg Malalasekera