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UNSTEADY VISCOUS FLOW. Viscous effects confined to within some finite area near the boundary → boundary layer. In unsteady viscous flows at low Re the boundary layer thickness δ grows with time; but in periodic flows, it remains constant. - PowerPoint PPT Presentation

Slide 1

UNSTEADY VISCOUS FLOW

Viscous effects confined to within some finite area near the boundary boundary layerIn unsteady viscous flows at low Re the boundary layer thickness grows with time; but in periodic flows, it remains constantIf the pressure gradient is zero, Navier-Stokes equation (in x) reduces to:

Heat Equation parabolic partial differential equation - linear Requires one initial condition and two boundary conditionsUy

Total of three conditionsImpulsively started plate

Stokes first problem

Heat Equation parabolic partial differential equation Can be solved by Separation of VariablesSuppose we have a solution:

Substituting in the diff eq:

May also be written as:

Moving variables to same side:

The two sides have to be equal for any choice of x and t ,

The minus sign in front of k is for convenienceThis equation contains a pair of ordinary differential equations:

increasing time

New independent variable:

is used to transform heat equation:

Substituting into heat equation:

Alternative solution toSeparation of Variables Similarity Solutionfrom:

To transform second order into first order:

2 BC turn into 1

With solution:

Integrating to obtain u:

Or in terms of the error function:

For > 2 the error function is nearly 1, so that u 0

For > 2 the error function is nearly 1, so that u 0Then, viscous effects are confined to the region < 2This is the boundary layer

grows as the squared root of timeincreasing time

UNSTEADY VISCOUS FLOWOscillating Plate

Ucos(t)yLook for a solution of the form:

Eulers formulaFouriers transform in the time domain:

B.C. in Y

Substitution into:

Most of the motion is confined to region within:

Ucos(t)y

UNSTEADY VISCOUS FLOWOscillating Plate

Look for a solution of the form:

Eulers formulaUcos(t)yWFouriers transform in the time domain:

B.C. in Y

Substitution into: