adam_flume_wall_model_library

ADAM, FLUME equilibrium wall model (issue #119, T3.4).

The thin-layer equations between the wall and a sample at height h, parallel velocity us, temperature ts, pressure ps (uniform across the layer): d/dy[(mu + mu_t) dU/dy] = 0, d/dy[(mu + mu_t) U dU/dy + cp (mu/Pr + mu_t/Pr_t) dT/dy] = 0, with mu_t = kappa y sqrt(rho tau_w) (1 - exp(-y+ / A+))^2, with the local density, y+ = y sqrt(tau_w rho_w) / mu_w, kappa 0.41, A+ 17, Pr_t 0.9 (Kawai & Larsson 2012). Both integrate once: (mu + mu_t) dU/dy = tau_w, cp (mu/Pr + mu_t/Pr_t) dT/dy + tau_w U = Q_w, Q_w = k_w dT/dy|_w. Given T(y), the two quadratures on a fixed stretched grid give tau_w and Q_w from the sample values: a fixed number of iterations, so a fixed cost per wall face on the device, with no tridiagonal solve.

Source: src/app/flume/common/adam_flume_wall_model_library.F90

Dependencies

Contents

• sample_cells
• wall_model_solve
• wall_model_solve_inverse
• wall_eddy_viscosity
• inverse_layer_speed
• inverse_strain
• log_mean_inverse

Variables

Name	Type	Attributes	Description
WM_POINTS	integer(kind=I4P)	parameter	Intervals of the layer; tau within 1 % up to h+ = 1e5, 5.5 % off at 1e6.
WM_ITERATIONS	integer(kind=I4P)	parameter	Iterations, Aitken every second one.
WM_STRETCH	real(kind=R8P)	parameter	Ratio of consecutive intervals.
WM_KAPPA	real(kind=R8P)	parameter	von Karman constant.
WM_APLUS	real(kind=R8P)	parameter	Damping constant.
WM_PRT	real(kind=R8P)	parameter	Turbulent Prandtl number.
WM_SRA	real(kind=R8P)	parameter	Reynolds analogy factor of the inverse model (Griffin et al. 2023, after Zhang et al. 2014).
WM_SHOTS	integer(kind=I4P)	parameter	Shots of the inverse model, secant on ln(tau) after the first.
WM_SUBITERATIONS	integer(kind=I4P)	parameter	Corrections of each step of the inverse model.

Subroutines

sample_cells

The cells of a wall-model sample: from the matching height hwm (0: the 3rd cell), the cell m counted from the wall and the weight w of cell m + 1, both within the n cells of the block, and the height of the sample actually taken.

Attributes: pure

subroutine sample_cells(hwm, dx, n, m, w, h)

Arguments

Name	Type	Intent	Attributes	Description
hwm	real(kind=R8P)	in		Matching height, spacing along the normal.
dx	real(kind=R8P)	in		Matching height, spacing along the normal.
n	integer(kind=I4P)	in		Cells of the block along the normal.
m	integer(kind=I4P)	out		Cell counted from the wall.
w	real(kind=R8P)	out		Weight of cell m + 1.
h	real(kind=R8P)	out		Height of the sample.

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wall_model_solve

Return the wall shear stress tau and the heat flux q (k dT/dy at the wall, positive when the gas is hotter than the wall) of an isothermal wall.

Attributes: pure

subroutine wall_model_solve(h, us, ts, ps, tw, R, cp, pr, mu0, tref, suth, omega_mu, tau, q, semilocal)

Arguments

Name	Type	Intent	Attributes	Description
h	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
us	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
ts	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
ps	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
tw	real(kind=R8P)	in		Wall temperature.
R	real(kind=R8P)	in		Gas constant, specific heat, Prandtl number.
cp	real(kind=R8P)	in		Gas constant, specific heat, Prandtl number.
pr	real(kind=R8P)	in		Gas constant, specific heat, Prandtl number.
mu0	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
tref	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
suth	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
omega_mu	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
tau	real(kind=R8P)	out		Wall shear stress, wall heat flux.
q	real(kind=R8P)	out		Wall shear stress, wall heat flux.
semilocal	logical	in	optional	Semi-local damping, y* = y sqrt(rho tau) / mu with the local rho, mu (else wall units).

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wall_model_solve_inverse

Return the wall shear stress tau and the heat flux q (k dT/dy at the wall, positive when the gas is hotter than the wall) of an isothermal wall by the model of Griffin, Fu & Moin (2023, JFM 970 A36): the strain of the constant-property layer at the same semi-local height, S_t = 1 / (1 + kappa y* D^2), D = 1 - exp(-y* / A+), y* = y sqrt(rho tau) / mu, mapped to the compressible one by the inverse total-stress-based transformation (their Eq. 2.2), with the temperature from the matching-anchored Reynolds analogy (Eq. 2.5) T = Tw + s Pr (Tr - Tw) (U / Ue) (1 - U / Us) + (U / Us)^2 (Ts - Tw), Tr = Te + Pr^(1/3) Ue^2 / (2 cp), s 1.14; so q = s cp tau (Tr - Tw) / Ue. Ue, Te: the state at the edge of the layer (the channel centre), given.

Eq. 2.2 written for the physical strain g = dU/dy, its property gradients through T(U): with E = (1 / S_t - 1) mu_w / mu, F = sqrt(rho / rho_w), K = tau F / mu, beta = (dT/dU) (1/2 + d ln mu / d ln T) / T (ideal gas at uniform pressure; the slope is omega_mu, less T / (T + S) under Sutherland), F beta y g^2 - (E + F + K beta y) g + K = 0; its positive root when beta < 0 (the other is negative), else the one nearer K / (E + F), the strain without the gradients (the exact tau / mu of the laminar layer, where the equation also holds the spurious root 1 / (beta y) of dy*/dy = 0). The layer is integrated from the wall on the stretched grid of wall_model_solve, each interval exact for 1 / g linear in y (the viscous and the log layer), corrected WM_SUBITERATIONS times, then shot on ln(tau) to the sample speed.

Attributes: pure

subroutine wall_model_solve_inverse(h, us, ts, ps, tw, ue, te, R, cp, pr, mu0, tref, suth, omega_mu, tau, q)

Arguments

Name	Type	Intent	Attributes	Description
h	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
us	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
ts	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
ps	real(kind=R8P)	in		Height, parallel speed, temperature and pressure of the sample.
tw	real(kind=R8P)	in		Wall temperature.
ue	real(kind=R8P)	in		Speed and temperature at the edge of the layer.
te	real(kind=R8P)	in		Speed and temperature at the edge of the layer.
R	real(kind=R8P)	in		Gas constant, specific heat, Prandtl number.
cp	real(kind=R8P)	in		Gas constant, specific heat, Prandtl number.
pr	real(kind=R8P)	in		Gas constant, specific heat, Prandtl number.
mu0	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
tref	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
suth	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
omega_mu	real(kind=R8P)	in		Viscosity law, as compute_dissipative_flux.
tau	real(kind=R8P)	out		Wall shear stress, wall heat flux.
q	real(kind=R8P)	out		Wall shear stress, wall heat flux.

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Functions

wall_eddy_viscosity

The eddy viscosity of the layer, kappa y sqrt(rho tau_w) D^2, D = 1 - exp(-y+ / A+) with y+ = y sqrt(tau_w rho_w) / mu_w (wall units) or, semilocal, y* = y sqrt(rho tau_w) / mu (local density and viscosity): the mu_t of wall_model_solve, at the height y of a cell below the matching height with that cell's density and viscosity (issue #119, wall_model_eddy_viscosity).

Attributes: pure

Returns: real(kind=R8P)

function wall_eddy_viscosity(y, rho, mu, tau, rhow, muw, semilocal) result(mt)

Arguments

Name	Type	Intent	Attributes	Description
y	real(kind=R8P)	in		Height, density and viscosity of the cell.
rho	real(kind=R8P)	in		Height, density and viscosity of the cell.
mu	real(kind=R8P)	in		Height, density and viscosity of the cell.
tau	real(kind=R8P)	in		Wall shear stress.
rhow	real(kind=R8P)	in		Density and viscosity at the wall.
muw	real(kind=R8P)	in		Density and viscosity at the wall.
semilocal	logical	in		Semi-local damping.

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inverse_layer_speed

The speed at the height h of the layer of wall stress tau of the inverse model (wall_model_solve_inverse).

Attributes: pure

Returns: real(kind=R8P)

function inverse_layer_speed(h, tau, us, ts, ps, tw, c1, R, mu0, tref, suth, omega_mu) result(u)

Arguments

Name	Type	Intent	Attributes	Description
h	real(kind=R8P)	in		Height, wall stress, sample speed, temperature, pressure, wall temperature.
tau	real(kind=R8P)	in		Height, wall stress, sample speed, temperature, pressure, wall temperature.
us	real(kind=R8P)	in		Height, wall stress, sample speed, temperature, pressure, wall temperature.
ts	real(kind=R8P)	in		Height, wall stress, sample speed, temperature, pressure, wall temperature.
ps	real(kind=R8P)	in		Height, wall stress, sample speed, temperature, pressure, wall temperature.
tw	real(kind=R8P)	in		Height, wall stress, sample speed, temperature, pressure, wall temperature.
c1	real(kind=R8P)	in		dT/dU at the wall.
R	real(kind=R8P)	in		Gas constant, viscosity law.
mu0	real(kind=R8P)	in		Gas constant, viscosity law.
tref	real(kind=R8P)	in		Gas constant, viscosity law.
suth	real(kind=R8P)	in		Gas constant, viscosity law.
omega_mu	real(kind=R8P)	in		Gas constant, viscosity law.

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inverse_strain

The strain dU/dy at the height y and speed u of the layer of wall stress tau of the inverse model (wall_model_solve_inverse).

Attributes: pure

Returns: real(kind=R8P)

function inverse_strain(y, u, tau, us, ts, ps, tw, c1, R, mu0, tref, suth, omega_mu) result(g)

Arguments

Name	Type	Intent	Attributes	Description
y	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
u	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
tau	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
us	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
ts	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
ps	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
tw	real(kind=R8P)	in		Height, speed, wall stress, sample speed, temperature, pressure, wall temperature.
c1	real(kind=R8P)	in		dT/dU at the wall.
R	real(kind=R8P)	in		Gas constant, viscosity law.
mu0	real(kind=R8P)	in		Gas constant, viscosity law.
tref	real(kind=R8P)	in		Gas constant, viscosity law.
suth	real(kind=R8P)	in		Gas constant, viscosity law.
omega_mu	real(kind=R8P)	in		Gas constant, viscosity law.

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log_mean_inverse

The mean of 1 / c(y) over an interval where c is linear between a and b.

Attributes: pure

Returns: real(kind=R8P)

function log_mean_inverse(a, b) result(r)

Arguments

Name	Type	Intent	Attributes	Description
a	real(kind=R8P)	in		End values.
b	real(kind=R8P)	in		End values.

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