Eulerian Fluid

3 Key excerpts on "Eulerian Fluid"

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  eBook - ePub

  Incompressible Flow

  • Ronald L. Panton(Author)
  • 2024(Publication Date)
  • Wiley

    (Publisher)

  Furthermore, if we employ velocity as our major dependent quantity instead of the particle position vector, we can usually find out all we want to know about a flow pattern. The Eulerian viewpoint is much more useful because physical laws written in terms of it do not contain the position vector, and the velocity appears as the major variable. It is interesting to note that a box of fluid does not change volume or rotate in the stagnation point flow. 4.2 EULERIAN VIEWPOINT The Eulerian viewpoint has us watch a fixed point in space as time proceeds. All flow properties, such as and are considered as functions of and. Assume the temperature of the fluid is given by. At a fixed time, tells how the temperature changes in space; at a fixed point, gives the local temperature history. The particle position vector in Eulerian variables is simply (4.8) The position vector in Eulerian variables has as components the local coordinates of the particle. Substituting into Eq. 4.2 and noting the obvious equivalence between the time variables, we have the transformation between Lagrangian and Eulerian variables as (4.9) These relations connect the Eulerian variables, and the Lagrangian variables. Particle path equations in Lagrangian variables are obtained by substituting in Eq. 4.9 and relegating to the role of an initial condition: (4.10) We retain as the function symbol in Eq. 4.10 to denote that this relation is a particle path function. (To be precise, recall that we used with two meanings in Eq. 4.2 : On the left-hand side it is the position vector, a dependent variable, while on the right-hand side it is the particle path function.) Streamlines in a flow are defined as lines that at any instant are tangent to the velocity vectors. If is a differential along a streamline, the tangency condition is expressed by the three equations (4.11) The form of Eq. 4.11 in vector calculus is (4.12) The cross-product of two nonzero vectors is zero only if they are parallel

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  eBook - ePub

  Automotive Aerodynamics

  • Joseph Katz(Author)
  • 2016(Publication Date)
  • Wiley

    (Publisher)

  Fig. 2.1 ), as viewed from a stationary frame of reference

  Figure 2.2 also demonstrates the principle of the Lagrangian formulation. In this case it is quite common to use a nonmoving, inertial frame of reference and to trace the dynamics of the particles (or group of particles) as shown in the figure. It is also clear that each group of particles must be followed individually, thereby significantly complicating the process. In contrast, the Eulerian approach observes a fixed control volume attached to the airfoil (as shown in Fig. 2.3 ), and as noted the streamlines and the whole problem appears to be independent of time. In most cases this results in a major simplification and therefore the Eulerian approach is widely used in classical fluid mechanics (and throughout this book).

  Figure 2.3

  Particle trajectories for the case shown in Fig. 2.2 , but as viewed in a control volume attached to the airfoil

  2.4 Pathlines, Streak Lines, and Streamlines

  Three sets of curves are normally associated with providing a pictorial description of a fluid motion; pathlines, streak lines, and streamlines. Pathlines: A curve describing the trajectory of a fluid element is called a pathline or a particle path. Pathlines are obtained in the Lagrangian approach by an integration of the equations of dynamics for each fluid particle. If the velocity field of a fluid motion is given in the Eulerian framework by Eq. 2.6 in a body-fixed frame, the pathline for a particle at P

  o

  in Fig. 2.1 can be obtained by an integration of the velocity. For steady flows the pathlines in the body-fixed frame become independent of time and can be drawn as in the case of flow over the airfoil shown in Fig. 2.1 and 2.3 . Streak lines: In many cases of experimental flow visualization, particles (e.g., dye or smoke) are introduced into the flow at a fixed point in space. The line connecting all of these particles is called a streak line. To construct streak lines using the Lagrangian approach, draw a series of pathlines for particles passing through a given point in space and, at a particular instant in time, connect the ends of these pathlines. Streamlines: Another set of curves can be obtained (at a given time) by lines that are parallel to the local velocity vector. To express analytically the equation of a streamline at a certain instant of time, at any point P in the fluid, the velocity must be parallel to the streamline element (Fig. 2.4

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  eBook - ePub

  Understanding Aerodynamics

  Arguing from the Real Physics

  • Doug McLean(Author)
  • 2012(Publication Date)
  • Wiley

    (Publisher)

  Because of the above general difficulties, analytic solutions to the NS equations are known for only a few simple cases with reduced dimensions and constant fluid properties, and even then only in limiting situations in which the inertia terms can be neglected. For example, there are effectively 1D solutions for steady, fully developed flow in planar 2D or circular-cross-section ducts or pipes and 2D solutions for flow around a circular cylinder or sphere in the limit of low Reynolds number. For some idealized situations at high Reynolds numbers, boundary-layer theory provides approximate solutions to the 2D NS equations that require only the solution of an ordinary differential equation (ODE) in 1D, as we'll see in Section 4.1. For more general flows, numerical solutions are our only option, unless we can make simplifying assumptions.

  3.4.6 The Physics as Viewed in the Eulerian Frame

  In the Eulerian description, we track what happens as fluid flows past points in a given spatial reference frame. So now, instead of tracking what happens to fixed parcels of fluid, as we did in the Lagrangian description, we track what happens in infinitesimal elements of volume imbedded in our spatial coordinate system. These Eulerian volume elements have fluid continuously streaming through them and across their bounding surfaces. This is, of course, the same streaming motion that was part of the flow when we described it in the Lagrangian frame. We are just seeing it now in a different reference frame, and the difference in vantage point requires us to treat the convection process differently when we implement our conservation laws. In the Lagrangian formulation, convection is accounted for implicitly by our definition of a fixed fluid parcel, and our conservation equations have no terms representing convection across the boundaries of a fluid parcel, because there is none by definition. In the Eulerian formulation, where there is generally a flux of fluid across the boundaries of our volume elements, the convection process must appear explicitly in the form of additional terms in the equations.

  Mathematically, the additional terms arise when we replace the time derivatives in the Lagrangian equations with their Eulerian equivalents, using Equation 3.2.1 . In the Eulerian equations that result, convection effects are represented by terms that arise from the term on the right-hand side. To see how this works, consider, for example, the x component of the momentum of a Lagrangian parcel of volume dV, which is given by ρ udV. Applying Equation 3.2.1 to this quantity gives

  3.4.1

  The second term on the right-hand side represents the convection of momentum in the Eulerian x-momentum equation in its rawest form. There is another form often seen in the literature, in which the density is taken outside the derivative, and the relationship to the Lagrangian acceleration Du/Dt is clearer. To derive this other form, we must invoke conservation of mass, which in its Lagrangian form simply states that the mass of a Lagrangian parcel doesn't change with time. Applying Equation 3.2.1

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