marenv::wake::LineKernelProfile Class Reference

#include <LineKernel.h>

Public Member Functions
	LineKernelProfile (const LineSource &source, const KernelParams &params)
double	XStart () const
	x_start = max(xStartFactor w, xStartMin), m.
const double *	Position () const
	Centre of the line, m, NED.
LineCoordinates	Coordinates (const double pos[3]) const
	Position of pos in the wake coordinates of the source.
KernelCentreline	At (double x) const
	Centreline amplitude A(x) and sigma at x >= x_start, m along flowDir; the amplitude is that of an infinitely long line (envelope 1).
double	Amplitude (double sigma, double envelope) const
	Centreline amplitude of the wake of sigma where the line carries the share envelope in [0, 1] of its momentum per unit length (the span envelope E(x, s)).
double	PeakAmplitude (double xFrom, double xTo) const
double	HalfSpan () const
	Projected half-length of the source, m.

Detailed Description

The kernel of LineDeficit() along one line source's wake.

With L the length of the source projected on the plane normal to the flow (length times |axis x flowDir|), w its width, m' = M / L the momentum deficit per unit length, C_T = min(M / (L w), 0.96), the kernel is a plane wake:

d(x, y, s) = A(x, s) exp(-y^2 / (2 sigma(x)^2)) for x >= x_start = max(xStartFactor w, xStartMin), else 0, with A the centreline amplitude below.

The width is that of the axisymmetric kernel with w for D: sigma = k x + epsilon w with the epsilon of Bastankhah and Porte-Agel (2014), up to x_f = max(farWakeStartFactor w, x_p), and beyond it the plane-wake similarity sigma = sigma_f (1 + 2 k (x - x_f) / sigma_f)^(1/2), so sigma^2 is linear in x and sigma and its slope are continuous at x_f; farWakeStartFactor = infinity keeps the linear growth. The centreline amplitude keeps the momentum of a Gaussian plane wake, integral d (1 - d) dy = m' / 2, which gives C(x) = (1 - sqrt(1 - m' / (sqrt(pi) sigma))) / sqrt(2), and C = 1 / sqrt(2), the largest value, where the radicand is negative (close to the source). A(x) = min(C(x), d_near) with the momentum-theory near wake d_near = 1 - sqrt(1 - C_T), so the amplitude does not rise with x. x_p is where C reaches min(d_near, 1 / sqrt(2)) (owner ruling R128, as C(x_p) = d_mom of the axisymmetric kernel): sigma_p = m' / (2 sqrt(pi) (sqrt(2) d - d^2)) with that d, which is m' / sqrt(pi) for C_T >= 0.914. Before x_p the amplitude is capped and the wake carries less than M, in proportion to sigma (as MENV-0030 for the axisymmetric kernel); from x_p on it carries all of it. Far downstream C ~ 1 / sigma ~ x^(-1/2).

Along the span the wake ends smoothly: E(x, s) = (erf((s + L / 2) / (sqrt(2) sigma)) - erf((s - L / 2) / (sqrt(2) sigma))) / 2, a box of the projected length convolved with a Gaussian of the same sigma, so integral E ds = L at every x. The line carries the share E of its momentum per unit length at s: d(x, y, s) = min(C(m' E, sigma), d_near) exp(-y^2 / (2 sigma^2)), so integral d (1 - d) dy = m' E / 2 at every s and the momentum through a plane, integral of that over s, is M / 2 from x_p on, where the amplitude is no longer capped. The wake spreads along the span as it does across it.

Constructor & Destructor Documentation

LineKernelProfile()

marenv::wake::LineKernelProfile::LineKernelProfile	(	const LineSource &	source,
		const KernelParams &	params
	)

Parameters

[in]	source	The line source; flowDir need not be normalised here.
[in]	params	The kernel parameters (expansion, xStartFactor, xStartMin, farWakeStartFactor).

Exceptions

std::invalid_argument

for a non-finite position, direction, length, width or momentum deficit, a non-positive length or width, a negative momentum deficit, a zero flowDir or axis, or an axis parallel to flowDir.

Member Function Documentation

PeakAmplitude()

double marenv::wake::LineKernelProfile::PeakAmplitude	(	double	xFrom,
		double	xTo
	)		const

Upper bound of At(x).amplitude for x in [xFrom, xTo], x_start <= xFrom <= xTo, m.

The amplitude does not rise with x (k >= 0, which Validate() requires of the fields), so the bound is its value at xFrom.

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The documentation for this class was generated from the following file:

• webfhsim/reloadrepos/marenv/include/marenv/wake/LineKernel.h