The standard $1-$equation model \ of turbulence was first derived by Prandtl and has evolved to be a common method for practical flow simulations. Five fundamental laws that any URANS model should satisfy are \[ \begin{ array} [c]{ ccc} \textbf{ 1.} & \text{ Time window:} & \begin{ array} [c]{ c} \tau\downarrow 0\text{ implies }v_{ \text{ { \small URANS}}}\rightarrow u_{ \text{ { \small NSE}}}\text{ \&}\\ \text{ }\tau\uparrow\text{ implies }\nu_{ T}\uparrow \end{ array} \\ \textbf{ 2.} & l(x)=0\ \text{ at walls:} & l(x)\rightarrow 0\text{ as }x\rightarrow walls,\\ \textbf{ 3.} & \text{ Bounded energy:} & \sup_{ t}\int\frac{ 1} { 2}|v(x,t)|^{ 2}+k(x,t)dx<\infty\\ \textbf{ 4.} & \begin{ array} [c]{ c} \text{ Statistical }\\ \text{ equilibrium:} \end{ array} & \lim\sup_{ T\rightarrow\infty}\frac{ 1}{ T}\int_{ 0}^{ T}\varepsilon _{ \text{ model}}(t)dt=\mathcal{ O}\left( \frac{ U^{ 3}}{ L}\right) \\ \textbf{ 5.} & \begin{ array} [c]{ c} \text{ Backscatter}\\ \text{ possible:} \end{ array} & \text{ (without negative viscosities)} \end{ array} \] This report proves that a \textit{ kinematic} specification of the model's turbulence lengthscale by \[ l(x,t)=\sqrt{ 2}k^{ 1/2}(x,t)\tau\text{ }, \] where $\tau$\ is the time filter window, results in a $1-$equation model satisfying Conditions 1,2,3,4 without model tweaks, adjustments or wall damping multipliers.
Categories: math.NA cs.NA