57 020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2007 Chapter 3 1 Chapter 3 Bernoulli Equation 3 1 Flow Patterns Streamlines Streaklines Pathlines A streamline is a line that is everywhere tangent to the velocity field Streamlines are obtained analytically by integrating the equations defining lines tangent to the velocity field For two dimensional flows the slope of the streamline dy dx must be equal to the tangent of the angle that the velocity vector makes with the x axis or dy v dx u If the velocity field is known as a function of x and y and t if the flow is unsteady this equation can be integrated to give the equation of the streamlines A pathline is the line traced out by a given particle as it flows from one point to another The pathline is a Lagrangian concept that can be produced in the laboratory by marking a fluid particle dying a small fluid element and taking a time exposure photograph of its motion Illustration of a pathline left and an example of pathlines motion of water induced by sur face waves right 57 020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2007 Chapter 3 2 A streakline consists of all particles in a flow that have previously passed through a common point Streaklines are more of a laboratory tool than an analytical tool They can be obtained by taking instantaneous photographs of marked particles that all passed through a given location in the flow field at some earlier time Such a line can be produced by continuously injecting marked fluid neutrally buoyant smoke in air or dye in water at a given location Illustration of a streakline left and an example of streaklines flow past a full sized streamlined vehicle in the GM aerodynamics laboratory wind tunnel and 18 ft by 34 ft test section facility driven by a 4000 hp 43 ft diameter fan right If the flow is steady pathlines streamlines and streaklines are the same For unsteady flows none of these three types of lines need be the same 3 2 Streamline Coordinates In the streamline coordinate system the flow is described in terms of one coordinate along the streamlines denoted s and the second coordinate normal to the streamlines denoted n Unit vectors in these two directions are denoted by s and n as shown in Fig 4 8 Of the major advantages of using the streamline coordinate system is that the velocity is always tangent to the s direction That is V v s s This allows simplifications in describing the fluid particle acceleration and in solving the equations governing the flow Chapter 3 3 57 020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2007 Figure 4 8 Streamline coordinate system for two dimensional flow For two dimensional flow we can determine the acceleration as a DV a s s an n Dt where a s and an are the streamline and normal acceleration respectively In general for steady flow both the speed and the flow direction are a function of location V V s n and s s s n Thus application of the chain rule gives a D v s s D v s D s s v s Dt Dt Dt or a v s v s ds v s dn s s ds s dn s v s t s dt n dt t s dt n dt This can be simplified by using the fact that the velocity along the streamline is v s ds dt and the particle remains on its streamline n constant so that dn 0 dt Hence 57 020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2007 Chapter 3 4 tv v vs s v st v ss a s s s s s Figure 4 9 Relationship between the unit vector along the streamline curvature of the streamline R s and the radius of The quantity s s represents the limit as s 0 of the change in the unit vector along the streamline s per change in distance along the streamline s The magnitude of s is constant s 1 it is a unit vector but its direction is variable if the streamlines are curved s s s s s s s s s s s s s s In the limit s 0 the direction of s s is seen to be normal to the streamline That is s s n n Thus by equating the above two equations of s s s n s s s can be expressed as s n n n s s R R 57 020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2007 Chapter 3 5 where s R from Fig 4 9 Similarly the quantity s t represents the limit as t 0 of the change in the unit vector along the streamline s per change in time t s s t t s t s s s t s t t t In the limit t 0 the direction of s t streamline That is is also seen to be normal to the s t n t Thus as t 0 s lim n n t t 0 t t Finally by multiplying v s with s t we get vs vn n n t t Note that the unit vector n is directed from the streamline toward the center of curvature Hence the acceleration for two dimensional flow can be written in terms of its streamwise and normal components in the form a 2 vs vn vs V vs s n t s t R or vs vs v n v 2s a s v s a t s n t R The first term v s t represents the local acceleration in streamline direction the second term v s v s s represents the convective acceleration along the streamline due to the spatial gradient i e convergence divergence streamlines 57 020 Mechanics of Fluids and Transport Processes Professor Fred Stern Fall 2007 Chapter 3 6 the third term v n t represents the local acceleration normal to flow and the last term v 2s R represents centrifugal acceleration one type of convective acceleration normal to the flow motion due to the streamline curvature 3 3 Euler Equation For an inviscid flow in which all the shearing stresses are zero 0 the general equations of motion See Chapter 6 reduce to g p a or g p V V V t These equations are commonly referred to as Euler s equations of motion named in honor of Leonhard Euler 1707 1783 a famous Swiss mathematician who pioneered work on the relationship between pressure and flow Although Euler s equations are considerably simpler than the general equations of motion they are still not amenable to a general analytical solution that would allow us to determine the pressure and velocity at all points within an inviscid flow field The main difficulty arises from the nonlinear velocity terms u u x v u x etc which appear in the convective acceleration Because of these terms Euler …
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