Fluids Lecture 17 Notes

Transcription

1 Flids Lectre 7 Notes. Obliqe Waves Readig: Aderso 9., 9. Obliqe Waves ach waves Small distrbaces created by a sleder body i a sersoic flow will roagate diagoally away as ach waves. These cosist of small isetroic variatios i, V,, ad h, ad are loosely aalogos to the water waves set ot by a seedboat. ach waves aear statioary with resect to the object geeratig them, bt whe viewed relative to the still air, they are i fact idistigishable from sod waves, ad their ormal-directio seed of roagatio is eqal to a, the seed of sod. sersoic flow still air V > a fixed body V body movig at sersoic seed statioary ach wave eqivalet fixed a movig ach wave (sod) The agle µ of a ach wave relative to the flow directio is called the ach agle. It ca be determied by cosiderig the wave to be the serositio of may lses emitted by the body, each oe rodcig a distrbace circle (i -D) or shere (i 3-D) which exads at the seed of sod a. At some time iterval t after the lse is emitted, the radis of the circle will be at, while the body will travel a distace V t. The ach agle is the see to be µ = arcsi at V t = arcsi which ca be defied at ay oit i the flow. I the sbsoic flow case where = V/a < the exadig circles do ot coalesce ito a wave frot, ad the ach agle is ot defied. V/a < V/a > µ at at Vt Vt

2 Obliqe shock ad exasio waves ach waves ca be either comressio waves ( > ) or exasio waves ( < ), bt i either case their stregth is by defiitio very small ( ). A body of fiite thickess, however, will geerate obliqe waves of fiite stregth, ad ow we mst distigish betwee comressio ad exasio tyes. The simlest body shae for geeratig sch waves is a cocave corer, which geerates a obliqe shock (comressio), or a covex corer, which geerates a exasio fa. The flow qatity chages across a obliqe shock are i the same directio as across a h obliqe shock h > h < o o o < > > > < h < o h < h = o o exasio fa ormal shock, ad across a exasio fa they are i the oosite directio. Oe imortat differece is that o decreases across the shock, while the fa is isetroic, so that it has o loss of total ressre, ad hece o = o. Obliqe geometry ad aalysis As with the ormal shock case, a cotrol volme aalysis is alied to the obliqe shock flow, sig the cotrol volme show i the figre. The to ad bottom bodaries are chose to lie alog streamlies so that oly the bodaries arallel to the shock, with area A, have mass flow across them. Velocity comoets are take i the x-z coordiates ormal ad tagetial to the shock, as show. The tagetial z axis is tilted from the stream flow directio by the wave agle, which is the same as the ach agle µ oly if the shock is extremely weak. For a fiite-stregth shock, > µ. The stream flow velocity comoets are = V si w = V cos z w V A A k i x V w ~ µ All the itegral coservatio eqatios are ow alied to the cotrol volme.

3 ass cotiity x-ometm z-ometm Eergy V ˆ da = 0 A + A = 0 = () V ˆ da + ˆ î da = 0 A + A A + A = 0 + = + () V ˆw da + ˆ ˆk da = 0 A w + A w = 0 V ˆh o da = 0 h o A + h o A = 0 w = w (3) h o = h o h + ( ) + w = h + ( + w) h + = h + (4) Eqatio of State = γ γ h (5) Simlificatio of eqatio (3) makes se of () to elimiate A from both sides. Simlificatio of eqatio (4) makes se of () to elimiate A ad the (3) to elimiate w from both sides. Obliqe/ormal shock eqivalece It is aaret that eqatios (), (), (4), (5) are i fact idetical to the ormal-shock eqatios derived earlier. The oe additio z-mometm eqatio (3) simly states that the tagetial velocity comoet does t chage across a shock. This ca be hysically iterreted if we examie the obliqe shock from the viewoit of a movig with the everywhere-costat tagetial velocity w = w = w. As show i the figre, the movig sees a ormal shock with velocities, ad. The static flid roerties,, h, a are of corse the same i both frames. Obliqe shock relatios The effective eqivalece betwee a obliqe ad a ormal shock allows re-se of the already derived ormal shock jm relatios. We oly eed to costrct the ecessary trasformatio from oe frame to the other. First we defie the ormal ach mber comoets see by the movig. a = V si a = si (6) a = V si( ) a = si( ) 3

4 w V V w chage frames of referece w fixed movig at w w = w = These are the related via or revios ormal-shock = f( ) relatio, if we make the sbstittios,. The fixed-frame the follows from geometry. = + γ γ γ = si( ) The static roerty ratios are likewise obtaied sig the revios ormal-shock relatios. (γ+) = (9) + (γ ) = + γ ( γ+ ) (0) h = () h o = ( ) γ/(γ ) h () o h To allow alicatio of the above relatios, we still reqire the wave agle. Usig the reslt w = w, the velocity triagles o the two sides of the shock ca be related by (7) (8) Solvig this for gives ta( ) ta = = = ta = ta (γ+) si + (γ ) si si (γ + cos ) + which is a imlicit defiitio of the fctio (, ). (3) Obliqe-shock aalysis: Smmary Startig from the kow stream ach mber ad the flow deflectio agle (body srface agle), the obliqe-shock aalysis roceeds as follows., Eq.(3) Eq.(6) Eqs.(7) (),,,, h h, o o Use of eqatio (3) i the first ste ca be roblematic, sice it mst be merically solved to obtai the (, ) reslt. A coveiet alterative is to obtai this reslt grahically, 4

5 from a obliqe shock chart, illstrated i the figre below. The chart also reveals a mber of imortat featres:. There is a maximm trig agle max for ay give stream ach mber. If the wall agle exceeds this, or > max, o obliqe shock is ossible. Istead, a detached shock forms ahead of the cocave corer. Sch a detached shock is i fact the same as a bow shock discssed earlier.. If < max, two distict obliqe shocks with two differet agles are hysically ossible. The smaller case is called a weak shock, ad is the oe most likely to occr i a tyical sersoic flow. The larger case is called a strog shock, ad is likely to form over a straight-wall wedge. The strog shock has a sbsoic flow behid it. 3. The strog-shock case i the limit 0 ad 90, i the er-left corer of the obliqe shock chart, corresods to the ormal-shock case. Μ strog shock Μ < 90 Μ weak shock Μ > Μ detached shock (bow shock) 30 (,Μ ) Μ > max max 5