<p>ME 362 Stream Function Page 1 of 3</p><p>Stream Function (  ):</p><p>   Definition: u  and v   y x</p><p>u v  Characteristic: It always satisfies the continuity equation (i.e.,   0 ) x y</p><p> Physical meaning:  = constant is a streamline</p><p>Change in  can be used to determine the flow direction</p><p>Stream Function (  ) vs. Volume Flow (Q) :</p><p>Consider two streamlines as shown below:</p><p> =  + d 2 1 dQ dy u  1</p><p> v</p><p> dx</p><p>Volume flow in = Volume flow out dQ  udy   vdx   u  , v     y x  dy  dx y x  d</p><p> 2 Q  d   dQ  d   2 1 1</p><p> Q is related to the change in </p><p> The flow direction can be determined by checking whether  increases or decreases ME 362 Stream Function Page 2 of 3</p><p>Flow Direction:</p><p>There are two ways to determine the direction of flow:</p><p>1) By looking at the change in :</p><p>         </p><p>Q  Q   </p><p> Q  2  1</p><p> If  2  1  Q is positive (indicates flow is to the right)</p><p> If  2  1  Q is negative (indicates flow is to the left)</p><p>2) By looking at direction of u and v:</p><p> streamlines</p><p> v V ~ Q</p><p> By the definition of streamline, the velocity field is always tangent to streamline u  Velocity (V) is related to volume flow rate (Q) (because Q = V*A)</p><p> We can find out the flow direction by looking at the velocity direction of u and v</p><p>Direction of u and v  Direction of V  Direction of Q ME 362 Stream Function Page 3 of 3</p><p>Solution Procedure: </p><p>Step 1: Ensure that continuity equation is satisfied</p><p>u v    0 x y</p><p>Step 2: Solve for stream function </p><p>  u     udy y  or   v       vdx x </p><p>Step 3: Find the flow direction</p><p> look at the change in </p><p> or</p><p> look at velocity direction of u and v</p><p>Example:</p><p>Given a 2-D velocity field:</p><p> y x u   ; v  ; r  x 2  y 2 r 2 r 2</p><p>1) Show that the flow field satisfies mass conservation 2) Find the corresponding stream function 3) Find the flow direction</p><p>To aid your analysis, here are some derivatives and integrals already performed for you: u 2xy u 2x 2 1    x r 4 y r 4 r 2 v 2y 2 1 v 2xy      x r 4 r 2 y r 4 y dy  ln x 2  y 2  f x  x 2  y 2</p>