Kutta condition

Different physical interpretations. Several important points must be emphasized when translating aerodynamics results into petroleum solutions. First, with respect to the preceding comments, the additional circulatory flow associated with the Kutta condition must be subtracted out before airfoil solutions can be applied to flows past impermeable shales. Second, not all aerodynamics solutions contain Kutta conditions the results for fractures derived in Chapter 5, for example, are taken from slender body crossflow theory where circulatory solutions are not needed. Third, in aerodynamics, the airfoil surface is a streamline of the flow having a constant value of the streamfunetion, supporting variable pressure in Darcy fracture flows, the fracture surface is not a streamline, but pressure is (or may be) constant along it. On the other hand, shale surfaces do represent streamlines, although Kutta s condition does not apply. Careful attention to the physics is obviously required. 

The higher order ODEs are reduced to systems of first-order equations and solved by the Runge-Kutta method. The missing condition at the initial point is estimated until the condition at the other end is satisfied. After two trials, linear interpolation is applied after three or more, Lagrange interpolation is applied. 

Exercise 1. From the values of Table 1 and Eq.(lO), write a computer program using a fourth order Runge-Kutta or fifth order Runge-Kutta-Fehlberg method and reproduce Figures 2, 3, 4, 5. In order to check that the chaotic behavior has been reached, it is necessary to run the program with two initial conditions very close, for example ... 

Equations (7) to (14) are solved numerically according to the Runge-Kutta method [145], with initial conditions as defined in equation (16). 

These Runge-Kutta methods do not require information from the past, and are very versatile if the time steps need to be adjusted as the solution evolves. The stability of the RK2 is similar to the APC2, while the RK4 has less strong conditions for stability [10]. Both are ideal for initial value problems in time or in space,... 

Equations 5-88, 5-89, and 5-90 are first order differential equations and the Runge-Kutta fourth order method with the boundary conditions is used to determine the concentrations versus time of the components. 

Since the problem is treated as an initial-value problem, one needs yo and yi before starting a four-step method. From the initial condition, yg = 0. The values y, i = 1,2,... are computed using the high order Runge-Kutta method of Prince and... 

In ref 164 new and elficient trigonometrically-fitted adapted Runge-Kutta-Nystrom methods for the numerical solution of perturbed oscillators are obtained. These methods combine the benefits of trigonometrically-fitted methods with adapted Runge-Kutta-Nystrom methods. The necessary and sufficient order conditions for these new methods are produced based on the linear-operator theory. 

In ref. 167 the preservation of some structure properties of the flow of differential systems by numerical exponentially fitted Runge-Kutta (EFRK) methods is investigated. The sufficient conditions on symplecticity of EFRK methods are presented. A family of symplectic EFRK two-stage methods with order four has been produced. This new method includes the symplectic EFRK method proposed by Van de Vyver and a collocation method at variable nodes that can be considered as the natural collocation extension of the classical RK Gauss method. 

Basic Theory. - 3.1.1 Explicit Runge-Kutta Form-Order Conditions. An i-stage explicit Runge-Kutta method used for the computation of the approximation of yn+i(x) in problem (1), when y (x) is known, can be expressed by the following relations ... 

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