Wind Energy Handbook. Michael Barton Graham

Wind Energy Handbook - Michael Barton Graham


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simple analytical formula for the tip‐loss function.

      The mathematical detail of Prandtl's analysis is given in Glauert (1935a), and because it is based on a somewhat strangely simplified model of the wake will not be repeated here. It has, however, remained the most commonly used tip‐loss correction because it is reasonably accurate and, unlike Goldstein's theory, the result can be expressed in closed solution form. The Prandtl tip‐loss factor is given by

      (3.80)f left-parenthesis r right-parenthesis equals StartFraction 2 Over pi EndFraction cosine Superscript negative 1 Baseline left-brace e Superscript minus pi left-parenthesis StartFraction upper R Super Subscript upper W Superscript minus r Over d EndFraction right-parenthesis Baseline right-brace

      Rwr is a distance measured from the wake edge. Distance d between the discs should be that of the distance travelled by the flow between successive vortex sheets. Glauert (1935a), takes d as being the normal distance between successive helicoidal vortex sheets.

      The helix angle of the vortex sheets ϕs is the flow angle assumed to be the same as ϕt, the helix angle at the blade tip, and so with B sheets intertwining from B blades and assuming that the discs move with the mean axial velocity in the wake, U(1 – a overbar):

      (3.81)d equals StartFraction 2 pi upper R Subscript w Baseline Over upper B EndFraction sine phi Subscript s Baseline equals StartFraction 2 pi upper R Subscript w Baseline Over upper B EndFraction StartFraction upper U Subscript infinity Baseline left-parenthesis 1 minus a overbar right-parenthesis Over upper W Subscript s Baseline EndFraction

Schematic illustration of Prandtl's wake-disc model to account for tip-losses. upper W equals StartRoot left-bracket upper U Subscript infinity Baseline left-parenthesis 1 minus a overbar right-parenthesis right-bracket squared plus left-parenthesis r upper Omega right-parenthesis squared EndRoot

      so

pi left-parenthesis StartFraction upper R minus r Over d EndFraction right-parenthesis equals StartFraction upper B Over 2 EndFraction left-parenthesis StartFraction upper R Subscript w Baseline minus r Over r EndFraction right-parenthesis StartRoot 1 plus StartFraction left-parenthesis r upper Omega right-parenthesis squared Over left-bracket upper U Subscript infinity Baseline left-parenthesis 1 minus a overbar right-parenthesis right-bracket squared EndFraction EndRoot

      and

      Although the physical basis of this model is not correct, it does quite effectively represent a convenient approximation to the attenuation towards the tips of the real velocities induced by the helicoidal vortex sheets.

      It should also be pointed out that the vortex theory of Figure 3.28 also predicts that the tip‐loss factor should be applied to the tangential flow induction factor.

      It is now useful to know what the variation of circulation along the blade is. For the previous analysis, which disregarded tip‐losses, the blade circulation was uniform [Eq. (3.69))].

italic rho upper W upper Gamma sine phi equals rho upper Gamma upper U Subscript infinity Baseline left-parenthesis 1 minus a Subscript b Baseline left-parenthesis r right-parenthesis right-parenthesis equals 4 pi rho StartFraction upper U Subscript infinity Baseline Superscript 3 Baseline Over upper Omega EndFraction ModifyingAbove a With bar left-parenthesis normal r right-parenthesis left-parenthesis 1 minus ModifyingAbove a With bar left-parenthesis normal r right-parenthesis right-parenthesis left-parenthesis 1 minus a Subscript b Baseline left-parenthesis r right-parenthesis right-parenthesis

      Recall that ab(r) is the flow factor local to the blade at radius r and Скачать книгу