Second Class of Simple Flows

1. Second Class of Simple Flows P M V Subbarao Professor Mechanical Engineering Department I I T Delhi It was not till Daniel Bernoulli published his celebrated treatise in the eighteenth century that the science of fluid Flow really began to get off the ground – G.I. Taylor Historically the First Fluid Flow Solution ….

2. Two Extreme Sides of Euler Equations Daniel Bernoulli’s chief work known as Hydrodynamica, published in 1738 An isentropic flow…. 1761: Euler’s Paper on General laws of the motion of fluids was published Mach's great contributions to understanding supersonic aerodynamics came as a revolutionary paper in 1887

3. Mach View of Fluid flows For an ideal gas: • Reference velocity : Velocity of sound in a selected fluid. • Velocity sound, c: For a real gas: For an incompressible liquids: Average bulk modulus for water is 2 X109 N/m2. In Mach’s view it is not possible to differentiate compressible and incompressible flows When flow Ma<0.2. Reduces the resolution of flow solutions….

5. Bernoulli’s View of Ideal Flow At time t+dt At time t A A’ B B’ Convert into a set of scar equations….. A Bernoulli’s flow field is a collection of stream lines.

6. Derivation of Bernoulli equation Consider Euler equation for incompressible flow: • The Bernoulli equation can be obtained as a scalar product of the Euler differential equation and a differential displacement vector. • For steady flow cases, the differential distance along the particle path is identical with a distance along a streamline. • Thus, the multiplication of Euler equation with the differential displacement , gives:

7. Scalar Equations to describe Ideal Flow field Use the definition of infinitesimal change in scalar variable. Integrating above Equation results in: Integrating above equation from a begin point B to an end point E:

8. Along a Stream line : From beginning to the end For an unsteady, incompressible flow, the integration of above equation delivers: For a steady, incompressible flow, the integration of above equation delivers: which is the Bernoulli equation. Valid for incompressible flows

9. The Greatest Natural Flow

10. Barotropic Atmosphere • The term barotropic is derived from the Greek baro, relating to pressure, and tropic, changing in a specific manner: • that is, in such a way that surfaces of constant pressure are coincident with surfaces of constant temperature or density. • A barotropic atmosphere is one in which the density depends only on the pressure, ρ = ρ(p). • The isobaric surfaces are also surfaces of constant density. • For an ideal gas, the isobaric surfaces will also be isothermal if the atmosphere is barotropic. Along a isobar : The geostrophic wind is independent of height in a barotropic atmosphere.

11. Geostrophic Wind Patterns Geostrophic wind is defined a horizontal velocity field,

12. Prediction of Simple Wind Patterns • The knowledge of the pressure distribution at any time determines the geostrophic wind. • This is true for large-scale motions away from the equator. • The geostrophic wind becomes the total horizontal velocity to within 10–15% in mid latitudes.

13. History of Fluid Dynamics Stokesian Kingdom Bernoulli Kingdom An experimental investigation of the circumstances which determine whether the motion of water in parallel channels shall be direct or sinuous and of the law of resistance in parallel channels

14. Honor of Osborne Reynolds : Engineering of Mathematics • Consider the Navier-Stokes equations with constant density it their dimensional form: Define dimensionless variables as: • Here U, L are assumed to be a velocity and length characteristic of the problem being studied. • In the case of flow past a body, L might be a body diameter and U the flow speed at infinity.

15. In low Mach number region with constant viscosity: Non-dimensionalization of NS Equation