Fluid Lab Lecture Note. Bernoulli Equation. Pitot-Static Tube 3. Aireed Meaurement 4. Preure Nondimenionalization Reference: Anderon 3.-3.5, Denker 3.4 ( htt://www.av8n.com/how/ ) Bernoulli Equation Definition For every oint along a treamline in frictionle flow with contant denity ρ, the local eed V () = V and local reure () are related by the Bernoulli Equation + ρ V = () Thi i a contant for all oint along the treamline, even though and V may vary. Thi i illutrated in the lot of () and () along a treamline near a wing, for intance. Standard terminology i a follow. = tatic reure ρv = dynamic reure = tagnation reure, or total reure Alo, a commonly-ued horthand for the dynamic reure i q ρv o we can alo write the Bernoulli equation in the following comact form. + q =
Uniform Utream Flow Cae Many ractical flow ituation have uniform flow omewhere utream, with V (x, y, z) = V and (x, y, z) = at every utream oint. Thi uniform flow can be either at ret with V 0 (a in a reervoir), or be moving with uniform velocity V 0 (a in a utream wind tunnel ection), a hown in the figure. V(x,y,z) (x,y,z) V(x,y,z) (x,y,z) V = 0 = cont. V = cont. = cont. Reervoir/Jet Wind Tunnel In thee ituation, i the ame for all treamline, and can be evaluated uing the utream condition in equation (). = = + ρv Thi give an alternative form of Bernoulli Equation which in thee flow ituation i valid for all downtream oint. + ρ V = (uniform utream flow) () If the utream quantitie are known, then i alo known, and thi equation then uniquely relate the reure field (x, y, z) and the eed field V (x, y, z). Retriction The Bernoulli Equation () i ubject to everal retriction. A tated above, one retriction i that the utream flow i uniform, o that and V are well-defined contant. Another retriction i that the treamline containing the oint where equation () i alied ha not been influenced by friction, or fluid vicoity, anywhere utream. Fortunately, thee vicoity-influenced oint are retricted to thin region called boundary layer, which are adjacent to olid urface a hown in the figure. In thee haded region the tagnation = o = o boundary layer reure ha ome unknown value which differ from the contant freetream value, (inide boundary layer)
o that the general Bernoulli Equation () i not uable to relate and V. Pitot-Static Tube A Pitot-Static tube i a device which allow meaurement of both the tatic and tagnation reure and at any oint in the flowfield. V, The ort i in the center of the front blunt end, which i faced into the oncoming flow. The flow tagnate (V dro to zero) at the ort location, o that locally = (at tagnation ort, where V = 0) which i then tranmitted by the ort tube to the reure-ening device. The tatic ort for i on the ide of the tream-aligned tube, o there i no local change in V, and o the local i tranmitted into the ort tube. Aireed Meaurement The main uroe of a itot-tatic tube i to meaure aireed. Uing the difference between the meaured and, the eed at the location of the itot-tatic tube can be comuted uing the Bernoulli Equation. (o ) V = ρ To allow afe oeration of an aircraft, it i neceary to meaure the flight eed. Thi i imly V, or the eed of the airma relative to the moving aircraft. Thi can be comuted from meaured and : (o ) V = ρ On an aircraft, meaurement of oe little difficulty. We know that (x, y, z) = 3
for any oint x, y, z which lie outide the thin boundary layer on the aircraft kin. So the itot tube can be ut into the flow almot anywhere on the aircraft to meaure the local and hence. In contrat, meauring the tatic reure i far more difficult, ince it i not oible to ut a fragile itot-tatic robe at infinity, or even far away from the aircraft reure field. The method actually ued i to lace everal tatic ort on the aircraft kin at trategic location, uch that the variou different local meaurement can be combined to rovide a ufficiently accurate etimate of the true freetream. Preure Nondimenionalization Preure Coefficient Force exerted by the fluid on an object tyically deend only on reure difference within the fluid rather than the abolute reure itelf. For examle, the lift on a wing deend on the reure difference between the to and bottom urface. At low aireed (low Mach number to be more recie), the abolute reure i very much greater than the reure difference of interet. Hence, it i convenient to bia and normalize the reure by uing and q, which define the dimenionle reure coefficient variable. C C = q (3) + 0 q The figure how how the axi i hifted and recaled into the C axi. Uing the Bernoulli Equation we can exre a = + q ρv which can be ued to eliminate from the C definition (3). Thi then give C in term of the eed alone. C = ρv = V (4) q So C i not only a normalized reure variable, but a normalized eed variable a well. V 4
Surface Preure Ditribution The mot convenient way to reent reure ditribution on a body uch a an airfoil i with C. The figure how the C ditribution on a NACA 44 airfoil for three angle of attack,, 3, and 5. The traditional reentation i with negative C uward, o that the tyically negative C f the to urface i on the tof the lot, and vice vera. 5