Partial Differential Equation based on their mathematical behavior can exhibit totally different solution 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 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 quasi-linear equation as shown above. u & v are the continuous function of x & y. We can imagine that u & v represents continuous velocity field in xy plane. At any given point in xy plane we have 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 exists) in xy plane that passes through point C along which derivatives of u & v are discontinuous or indeterminant.
The existence of these Characteristics Lines are identifiable curve in the xy plane. Therefore we should be able to calculate equation for these curves, and specially the slope of these curves through that 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 gives the slopes of lines along which derivative 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 exists through point C in XY plane. Such equations are known as Hyperbolic equation.
If D = 0 – Such equations are called Parabolic equation.
if D < 0 – The characteristics curves are imaginary and equations are called Elliptical equation.
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 eigen value of matrix [N] determines the classification of PDE. If the eigen value are all real then the equations are hyperbolic, If the eigen value are complex then the equations are elliptical.
Consider 2D irrotational, 2D, inviscid, steady flow of a compressible gas. If the flow field is slightly perturbed from its freestream conditions, may be 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 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 System of equations, the eigen value could be complex of both real and complex values. Such equations exhibit Mixed Elliptical-Hyperbolic behavior equation. Full Naiver Stokes Equation are the one such example.
Characteristics of PDEs are very important to understand from the mathematical and physical perspective. It is very important to understand how mathematical characteristic of PDE reflects the physics. Each type of equation has a different mathematical behavior and this reflect 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 equation with 2 characteristic lines that defines 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. Solution of hyperbolic Equations is set up as 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 the example of such 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 problem.
Hyperbolic equations with two characteristics across which we can have possible discontinuities are valid for super-sonic flows. Disturbance propagate in all directions at 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 source is moving much faster than disturbance. So disturbance dissipation is insignificant that 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’s lead to 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 characteristic 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 the 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 initial condition solution is marched either in time or space. Parabolic Equations lend themselves to marching type solution, analogous to hyperbolic equations.
Steady Boundary Layer flow is example for parabolic equation. When the Reynolds number based on body length is large and assuming 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 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 with in boundary layer simplified Boundary Layer equations can be solved. As shown in above picture of Boundary Layer flow on an aerofoil, boundary condition comes from the aerofoil surface “no-slip condition” and outer edge of boundary layer where “inviscid flow conditions” are defined. We need to define initial data of flow field may be near the leading edge of 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 condition, we march forward along the aerofoil direction to compute the solution in boundary layer.
Unsteady heat conduction problem is governed by Parabolic equation i.e. parabolic in time. Assume we have fluid separated by two parallel plate, Initially fluid and plate temperature are same and the system is in equilibrium. Then at t = 0, Temperature of right plate is impulsively increased, Temperature field of that instant is the initial condition and temperature of left and right plate are 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 the all directions with infinite speed. Parabolic Equation with time marching problem has one discontinuity. We have an initial condition. Discontinuity felt in domain as time marches because disturbance is imposed at the boundary of the domain propagates in all directions with infinite speed. Due to the significant amount of dissipation, the system will try to homogenies the disturbance in the domain that is imposed at the boundary.
The characteristics curve for Elliptical Equation are imaginary. We don’t have limited region of influence or domain of dependence, rather information is propagated everywhere in all directions. This is the major characteristics of Elliptical equation. Let us consider a closed domain abcd as shown in 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 solution approach is totally different from the “marching approach” of parabolic and hyperbolic equations. We need to specify boundary condition for all boundaries in the computational domain. These boundary condition can be Dirichlet or Newmann or Mixed type. So this is a Boundary-Value Problem.
Steady Inviscid Subsonic Flow are 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 aerofoil deflected upward and behind the aerofoil deflected downward. Disturbance introduced by the presence of aerofoil are felt throughout the entire flow field. Above picture is a picture consistent with the mathematical behavior of elliptical equation.
Steady state conduction is an example of Elliptical equation. In the elliptical equation disturbance propagates in all directions at infinite speed. Solution in domain always smooth/continuous because disturbance propagates in all directions at infinite speed. (Boundary value problem) Boundary can have discontinuities so the effect of discontinuities from the boundary dissipates very rapidly & Disturbance felt and dissipates rapidly in the entire domain.
Above table summarize 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 form 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. Full Navier-Stokes equation exhibits mixed mathematical behavior. Therefore when we talk about what type of fluid flow field are governed by Parabolic equations, we are really talking about what types of approximated flow-field models are governed by Parabolic equations. Same philosophy is followed for Hyperbolic and Elliptical equations. Full Navier-Stokes equation have 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. Time-Marching approach is a well-posed, most numerical solution to the full, compressible Navier Stokes equations uses “time marching approach”.
Reference: Chapter-3 Computational Fluid Dynamics by John D. Anderson, Chapter-2 An introduction to CFD by Versteeg Malalasekera