  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.
