Solution of Inverse Mixed Boundary Value Problem of Aerohydrodynamics for Cascade of Profiles without Additional Restric
- PDF / 163,472 Bytes
- 6 Pages / 612 x 792 pts (letter) Page_size
- 31 Downloads / 187 Views
IEF COMMUNICATIONS
Solution of Inverse Mixed Boundary Value Problem of Aerohydrodynamics for Cascade of Profiles without Additional Restrictions R. B. Salimov1* 1
Kazan State Architecture and Building University, 1 Zelenaya str., Kazan, 420043 Russia
Received April 11, 2020; revised April 11, 2020; accepted June 29, 2020
Abstract—We consider an inverse boundary-value problem for a aerohydrodynamical cascade of profiles streamlined by a potential flow of an incompressible inviscid fluid. It is required to find the shape of the profile of the cascade, by a given velocity distribution as a function of the arc abscissa, and determine the period of the cascade. DOI: 10.3103/S1066369X20080083 Key words: inverse boundary value problem of aerohydrodynamics, grill, profile, complex potential.
1. Let we have a cascade of profiles in the plane of complex variable z = z + iy with period teiν (see [1, 2]), t > 0, 0 < ν < π. The cascade is streamlined by a steady flow of inviscid fluid; denote by w(z) its complex potential. Denote by ν1 , e−iα1 − π2 < α1 < π2 , and ν2 e−iα2 , − π2 < α2 < π2 , the values of complex velocity w (z) at infinity from the left and from the right of the cascade. The flow has period teiν , i.e. w (z + teiν ) = w (z). Denote by Lz the profile of the cascade, the branch-point on which coincides with z = 0. Let us introduce the plane of complex variable ζ = ρeiγ (ρ > 0, Im γ = 0). Let z = z(ζ) map conformally the exterior of the cascade onto an infinite-sheeted Riemann surface lying in the interior of a system of circles |ζ| = 1. In addition, we claim that infinite points, located on the left and on the right of the cascade, match to the points ζ = −q and ζ = q, 0 < q < 1. The function maps conformally the disk |ζ| < 1 with the slit along the segments with endpoints ζ = −q, ζ = q, onto the domain bounded with the profile of the cascade and two congruent lines; the difference between complex coordinates of the corresponding points of the lines lying above and below the profiles is equal to teiν (see [1]). Further we will consider the branch of z(ζ) which maps the circle onto the profile Lz . Denote w(z(ζ)) = W (ζ);
(1)
we will consider W (ζ) as the complex potential of the corresponding flow in the domain |ζ| ≤ 1. For the derivative of W (ζ), we have the representation [1] γA +γB 1 −1 dW = iB1 e−i 2 (ζ − eiγA )(ζ − eiγB )(ζ 2 − q 2 )−1 ζ 2 − 2 (2) dζ q *
E-mail: [email protected]
64
SOLUTION OF INVERSE MIXED BOUNDARY VALUE PROBLEM
65
where B1 is a real number, B1 = 0. Besides, eiγA and eiγB are the points on the circle |ζ| = 1, matching to the leading edge A and the trailing edge B of the flow lying on the profile Lz of the cascade. We will fix the direction of the bypass of Lz and |ζ| = 1 such that the flow domain is on the left while traveling the curves. The circulation Γ of velocity along the circle |ζ| = 1 with the above direction of bypass is [1]: dW dζ, (3) Γ= |ζ|=1 dζ the same is the circulation of velocity along Lz . Taking into account formulas (2) and (3) we obtain γA − γB . (4) 2 The deri
Data Loading...
