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Chapter 2. Flows with Rotating Reference Frames

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Chapter 2. Flows with Rotating Reference Frames Chapter 2. Flows with Rotating Reference Frames This chapter describes the theoretical background modeling flows in rotating reference frames. The information in this chapter is presented in the following sections: • Section 2.1: Introduction • Section 2.2: Flow in a Rotating Reference Frame • Section 2.3: Flow in Multiple Rotating Reference Frames 2.1 Introduction ANSYS FLUENT solves the equations of fluid flow and heat transfer, by default, in a stationary (or inertial) reference frame. However, there are many problems where it is advantageous to solve the equations in a moving (or non-inertial) reference frame. Such problems typically involve moving parts (such as rotating blades, impellers, and similar types of moving surfaces), and it is the flow around these moving parts that is of interest. In most cases, the moving parts render the problem unsteady when viewed from the stationary frame. With a moving reference frame, however, the flow around the moving part can (with certain restrictions) be modeled as a steady-state problem with respect to the moving frame. ANSYS FLUENT’s moving reference frame modeling capability allows you to model prob- lems involving moving parts by allowing you to activate moving reference frames in selected cell zones. When a moving reference frame is activated, the equations of mo- tion are modified to incorporate the additional acceleration terms which occur due to the transformation from the stationary to the moving reference frame. By solving these equations in a steady-state manner, the flow around the moving parts can be modeled. For many problems, it may be possible to refer the entire computational domain to a single moving reference frame. This is known as the single reference frame (or SRF) approach. The use of the SRF approach is possible; provided the geometry meets certain requirements (as discussed in Section 2.2: Flow in a Rotating Reference Frame). For more complex geometries, it may not be possible to use a single reference frame. In such cases, you must break up the problem into multiple cell zones, with well-defined interfaces between the zones. The manner in which the interfaces are treated leads to two approximate, steady-state modeling methods for this class of problem: the multiple reference frame (or MRF) approach, and the mixing plane approach. These approaches Release 12.0 c ANSYS, Inc. January 29, 2009 2-1 Flows with Rotating Reference Frames will be discussed in Sections 2.3.1 and 2.3.2. If unsteady interaction between the sta- tionary and moving parts is important, you can employ the Sliding Mesh approach to capture the transient behavior of the flow. The sliding meshing model will be discussed in Chapter 3: Flows Using Sliding and Deforming Meshes. outlet periodic boundaries inlet Figure 2.1.1: Single Component (Blower Wheel Blade Passage) 2.2 Flow in a Rotating Reference Frame The principal reason for employing a moving reference frame is to render a problem which is unsteady in the stationary (inertial) frame steady with respect to the moving frame. For a steadily rotating frame (i.e., the rotational speed is constant), it is possible to transform the equations of fluid motion to the rotating frame such that steady-state solutions are possible. By default, ANSYS FLUENT permits the activation of a moving reference frame with a steady rotational speed. If the rotational speed is not constant, the transformed equations will contain additional terms which are not included in ANSYS FLUENT’s formulation (although they can be added as source terms using user-defined functions). It should also be noted that you can run an unsteady simulation in a moving reference frame with constant rotational speed. This would be necessary if you wanted to simulate, for example, vortex shedding from a rotating fan blade. The unsteadiness in this case is due to a natural fluid instability (vortex generation) rather than induced from interaction with a stationary component. The information in this section is presented in the following: • Section 2.2.1: Equations for a Rotating Reference Frame • Section 2.2.2: Single Rotating Reference Frame (SRF) Modeling 2-2 Release 12.0 c ANSYS, Inc. January 29, 2009 2.2 Flow in a Rotating Reference Frame stationary zone interface rotating zone Figure 2.1.2: Multiple Component (Blower Wheel and Casing) 2.2.1 Equations for a Rotating Reference Frame Consider a coordinate system which is rotating steadily with angular velocity ~ω relative to a stationary (inertial) reference frame, as illustrated in Figure 2.2.1. The origin of the rotating system is located by a position vector ~r0. The axis of rotation is defined by a unit direction vectora ˆ such that ~ω = ωaˆ (2.2-1) The computational domain for the CFD problem is defined with respect to the rotating frame such that an arbitrary point in the CFD domain is located by a position vector ~r from the origin of the rotating frame. The fluid velocities can be transformed from the stationary frame to the rotating frame using the following relation: ~vr = ~v − ~ur (2.2-2) where ~ur = ~ω × ~r (2.2-3) Release 12.0 c ANSYS, Inc. January 29, 2009 2-3 Flows with Rotating Reference Frames Figure 2.2.1: Stationary and Rotating Reference Frames In the above, ~vr is the relative velocity (the velocity viewed from the rotating frame), ~v is the absolute velocity (the velocity viewed from the stationary frame), and ~ur is the “whirl” velocity (the velocity due to the moving frame). When the equations of motion are solved in the rotating reference frame, the accel- eration of the fluid is augmented by additional terms that appear in the momentum equations [17]. Moreover, the equations can be formulated in two different ways: • Expressing the momentum equations using the relative velocities as dependent vari- ables (known as the relative velocity formulation). • Expressing the momentum equations using the absolute velocities as dependent variables in the momentum equations (known as the absolute velocity formulation). The exact forms of the governing equations for these two formulations will be provided in the sections below. It can be noted here that ANSYS FLUENT’s pressure-based solvers provide the option to use either of these two formulations, whereas the density-based solvers always use the absolute velocity formulation. For more information about the advantages of each velocity formulation, see Section 10.7.1: Choosing the Relative or Absolute Velocity Formulation (in the separate User’s Guide). 2-4 Release 12.0 c ANSYS, Inc. January 29, 2009 2.2 Flow in a Rotating Reference Frame Relative Velocity Formulation For the relative velocity formulation, the governing equations of fluid flow for a steadily rotating frame can be written as follows: Conservation of mass: ∂ρ + ∇ · ρ~v = 0 (2.2-4) ∂t r Conservation of momentum: ∂ (ρ~v ) + ∇ · (ρ~v ~v ) + ρ(2~ω × ~v + ~ω × ~ω × ~r) = −∇p + ∇ · τ + F~ (2.2-5) ∂t r r r r r Conservation of energy: ∂ (ρE ) + ∇ · (ρ~v H ) = ∇ · (k∇T + τ · ~v ) + S (2.2-6) ∂t r r r r r h The momentum equation contains two additional acceleration terms: the Coriolis acceler- ation (2~ω×~vr), and the centripetal acceleration (~ω×~ω×~r). In addition, the viscous stress (τ r) is identical to Equation 1.2-4 except that relative velocity derivatives are used. The energy equation is written in terms of the relative internal energy (Er) and the relative total enthalpy (Hr), also known as the rothalpy. These variables are defined as: p 1 E = h − + (v 2 − u 2) (2.2-7) r ρ 2 r r p H = E + (2.2-8) r r ρ Absolute Velocity Formulation For the absolute velocity formulation, the governing equations of fluid flow for a steadily rotating frame can be written as follows: Conservation of mass: ∂ρ + ∇ · ρ~v = 0 (2.2-9) ∂t r Conservation of momentum: ∂ ρ~v + ∇ · (ρ~v ~v) + ρ(~ω × ~v) = −∇p + ∇ · τ + F~ (2.2-10) ∂t r Release 12.0 c ANSYS, Inc. January 29, 2009 2-5 Flows with Rotating Reference Frames Conservation of energy: ∂ ρE + ∇ · (ρ~v H + p~u ) = ∇ · (k∇T + τ · ~v) + S (2.2-11) ∂t r r h In this formulation, the Coriolis and centripetal accelerations can be collapsed into a single term (~ω × ~v). 2.2.2 Single Rotating Reference Frame (SRF) Modeling Many problems permit the entire computational domain to be referred to a single rotating reference frame (hence the name SRF modeling). In such cases, the equations given in Section 2.2.1: Equations for a Rotating Reference Frame are solved in all fluid cell zones. Steady-state solutions are possible in SRF models provided suitable boundary conditions are prescribed. In particular, wall boundaries must adhere to the following requirements: • Any walls which are moving with the reference frame can assume any shape. An example would be the blade surfaces associated with a pump impeller. The no slip condition is defined in the relative frame such that the relative velocity is zero on the moving walls. • Walls can be defined which are non-moving with respect to the stationary coor- dinate system, but these walls must be surfaces of revolution about the axis of rotation. Here the so slip condition is defined such that the absolute velocity is zero on the walls. An example of this type of boundary would be a cylindrical wind tunnel wall which surrounds a rotating propeller. Rotationally periodic boundaries may also be used, but the surface must be periodic about the axis of rotation. As an example, it is very common to model flow through a blade row on a turbomachine by assuming the flow to be rotationally periodic and using a periodic domain about a single blade.
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