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Original publisher: [Washington, D.C.] : National Aeronautics and Space Administration, Office of Management, Scientific and Technical Information Division, 1991. OCLC Number: ocm27903759 Subject: Aerodynamics. Excerpt: ... # v ( k ) = $ _ h ( 7 ) eik77d7 ( 3.8 )-ik77. 7 ( 3 9 ) h ( 7 ) = ___ _v ( kT ) e dK Inserting Equation 3.9 into Equation 3.7 and performing the time derivative gives 2-ik7 ( 7 + Ut ) dk 8q P ° U 6 ( _ + FA ) fk_ _v ( kT ) e ( 3.10 ) Po _ = " 2_ " Y To express Equation 3.10 in cylindrical coordinates, consider the r = constant surface to be unwrapped onto a plane. Then the 7, f system can be rotated into x, 4 using the relations ( 3.11 ) _r r4 7--uV-x-_ ( 3.12 ) _r V r_ _ = vX-u where the coefficients V / U and _r / U are tied to the advance triangle shown in Figure 4. The inverses of these equations are shown for future reference r @ _r V ( 3 lla ) = _-0-7-U_ Uv7 x =-+ _ ( 3.12a ) For the delta function in Equation 3.10, we first note that it is even with respect to argument and then substitute Equation 3.12 for 6 ( _ + FA ) = 6 [ _ ( _-_V---v_FA ) ] = __ _x U U 6 ( 4-FA ) ( 3.13 ) Vr V Vr The second line in Equation 3.13 follows from the integral properties of any func-tion, say f ( ax ): ff ( ax ) dx = i / a 7f ( x ) dx. Now, with the substitution of Equation 3. ii for 7, the thickness source has the form 3 _x U aq-P ° U 6 ( _-FA ) Po at 2_Vr V Vr v _ + _., ikT ( uX urn ) e-ikTUtdk ( 3 14 ) × fk_ v ( k7 ) e 7 ' But the source exists only on _ = C:x / V + ( U / Vr ) FA so that this can be substituted for _ in the exponential. The result, after recognizing that U z = V 2 + _Zr2 ( 3.15 ) defining a = U / V ( 3.16 ) and ( 3.17 ) Kx = ak_ and expanding the delta function via ein = ( 3.18 ) I'X =-CO-9-
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Unified aeroacoustics analysis for high speed turboprop aerodynamics and noise
Bokrecensioner » Unified aeroacoustics analysis for high speed turboprop aerodynamics and noise
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