|
The vorticity equation of fluid dynamics describes evolution of the vorticity of a particle of a fluid as it moves with its flow, that is, the local rotation of the fluid (in terms of vector calculus this is the curl of the flow velocity). The equation is: : where ''D''/''Dt'' is the material derivative operator, also denoted by in capital D notation as ''D''/''Dt'', is the flow velocity, ''ρ'' is the local fluid density, ''p'' is the local pressure, ''τ'' is the viscous stress tensor and represents the sum of the external body forces. The first source term on the right hand side represents vortex stretching. The equation is valid in the absence of any concentrated torques and line forces, for a compressible Newtonian fluid. In the case of incompressible (i.e. low Mach number) and isotropic fluids, with conservative body forces, the equation simplifies to the vorticity transport equation : where ''ν'' is the kinematic viscosity and ∇2 is the Laplace operator. ==Physical Interpretation== * The term ''D/Dt'' on the left-hand side is the material derivative of the vorticity vector . It describes the rate of change of vorticity of the fluid particle. This change can be attributed to unsteadiness in the flow (∂/∂''t'', the ''unsteady term'') or due to the motion of the fluid particle as it moves from one point to another ( ∙ (), the ''convection term''). * The term (∙ ) on the right-hand side describes the stretching or tilting of vorticity due to the flow velocity gradients. Note that is an order-2 tensor with nine components. * The term ( ∙ ) describes stretching of vorticity due to flow compressibility. It follows from the Navier-Stokes equation for continuity,namely :: :or :: :where ''v'' = 1/''ρ'' is the specific volume of the fluid element. One can think of ∙ as a measure of flow compressibility. Sometimes the negative sign is included in the term. * The term (1/''ρ''2)''ρ'' × ''p'' is the baroclinic term. It accounts for the changes in the vorticity due to the intersection of density and pressure surfaces. * The term × ( ∙ ''τ''/''ρ''), accounts for the diffusion of vorticity due to the viscous effects. * The term × provides for changes due to external body forces. These are forces that are spread over a three-dimensional region of the fluid, such as gravity or electromagnetic forces. (As opposed to forces that act only over a surface (like drag on a wall) or a line (like surface tension around a meniscus). 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「vorticity equation」の詳細全文を読む スポンサード リンク
|