No-slip continuum CFD observable error

Assume Countable Choice. Let $D^{\ast }\subset {\mathbb{R}}^3$ be a nonempty bounded connected polyhedral open set in dimensionless coordinates whose boundary has a finite rigidly moved Lipschitz graph-chart atlas, and let $T^{\ast }>0$ be dimensionless final time. Choose ${\mathrm{\ell }}_0>0$ metres and $U_0>0$ metres per second, so the physical domain is ${\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mathrm{\ell }}_0{D}^{\ast }$ and physical time is $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=({\mathrm{\ell }}_0\mathrm{/}{U}_0)t^{\ast }$. Let constant density ${\rho }_0>0$ have units $\mathrm{k}\mathrm{g}{\mathrm{m}}^{-3}$ and constant dynamic shear viscosity ${\mu }_0>0$ have units $\mathrm{P}\mathrm{a}\mathrm{s}$. Put $I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=(0,({\mathrm{\ell }}_0\mathrm{/}{U}_0)T^{\ast })$. Throughout this item use the explicitly reindexed physical fields $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,t):=u(t,x)$, $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,t):=p(t,x)$, and $b_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,t):=b(t,x)$, where the right-hand fields have the time-first domain $I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times {\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$ of the physical model definitions; the same notation convention applies to their explicitly adopted distributional weak extension below. Thus $u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to {\mathbb{R}}^3$ and $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to \mathbb{R}$. Let $b_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\times I_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\to {\mathbb{R}}^3$ be continuous, measured in $\mathrm{m}{\mathrm{s}}^{-2}$. Set $u^{\ast }(y,t^{\ast }):=U_0^{-1}{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y,{\mathrm{\ell }}_0{t}^{\ast }\mathrm{/}{U}_0),p^{\ast }(y,t^{\ast }):=({\rho }_0{U}_0^2{)}^{-1}{p}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y,{\mathrm{\ell }}_0{t}^{\ast }\mathrm{/}{U}_0),f^{\ast }(y,t^{\ast }):={\mathrm{\ell }}_0{U}_0^{-2}{b}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}({\mathrm{\ell }}_{0}y,{\mathrm{\ell }}_0{t}^{\ast }\mathrm{/}{U}_0),{\nu }^{\ast }:=\frac{{\mu }_0}{{\rho }_0{U}_0{\mathrm{\ell }}_0}>0.$ Thus $u^{\ast },p^{\ast },f^{\ast },{\nu }^{\ast }$, the mesh size $h$, and all time steps ${\tau }_n$ in the mathematical estimates are dimensionless; the dimensional kinematic viscosity is ${\nu }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\mu }_0\mathrm{/}{\rho }_0$ in ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, and the Reynolds number at these scales is $\mathrm{Re}={\rho }_0{U}_0{\mathrm{\ell }}_0\mathrm{/}{\mu }_0=1\mathrm{/}{\nu }^{\ast }$. Adopt the forced constant-density incompressible Newtonian branch and the stationary no-slip boundary choice from the preceding model definitions: $\rho ={\rho }_0$, $\mathrm{div}{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=0$, ${\sigma }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=-p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}{I}_3+2{\mu }_0{D}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, and ${\rho }_0({\mathrm{\partial }}_t{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+(u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}\cdot \mathrm{\nabla })u_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}})={\rho }_0{b}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+\mathrm{div}{\sigma }_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}$, where $D_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}=({\mathrm{\nabla }}_x{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}+{\mathrm{\nabla }}_x{u}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}^{\mathsf{T}})\mathrm{/}2$ and stress divergence is rowwise. The pressure is a constraint multiplier enforcing incompressibility, not an equation-of-state pressure. The selected no-slip condition is zero velocity trace on the stationary wall. The dimensionless pressure uses the spatial mean-zero gauge.

Use the complete smooth MINI observation interface defined earlier on the incompressible-numerical page. In particular, $V=H_0^1(D^{\ast }{)}^3$ has norm $\mathrm{\parallel }v{\mathrm{\parallel }}_V=\mathrm{\parallel }\mathrm{\nabla }v{\mathrm{\parallel }}_{L^2(D^{\ast })}$, $Q=L_0^2(D^{\ast })$ has its $L^2$ norm, and $V_{\sigma }=\{v\in V:\mathrm{div}v=0\}$. The velocity space is continuous piecewise affine enriched by one interior simplex bubble per component, the pressure space is continuous piecewise affine with zero mean, and the discrete velocity belongs to the weak divergence kernel $Z_h$. Let $0=t_0<\cdots <t_N=T^{\ast }$ be the finite dimensionless time partition, ${\tau }_n=t_n-t_{n-1}>0$, ${\tau }_{\max }={\max }_n{\tau }_n$, and $h={\max }_{K\in {\mathcal{T}}_h}\mathrm{diam}(K)$ on a conforming shape-regular affine simplicial mesh ${\mathcal{T}}_h$. Forcing is averaged over each interval, the initial discrete velocity is $u_h^0=P_{Z_h}{u}^{\ast }(0)$, and each step uses the stated mixed backward-Euler recurrence with the real skew convective form; choose any actual discrete root. Assume $f^{\ast }\in L^2(0,T^{\ast };L^2(D^{\ast }{)}^3)$, the exact solution satisfies $u^{\ast }\in W^{1,\mathrm{\infty }}(0,T^{\ast };H^2(D^{\ast }{)}^3\cap V_{\sigma })$ and $p^{\ast }\in W^{1,\mathrm{\infty }}(0,T^{\ast };H^1(D^{\ast })\cap Q)$, and the exact and discrete fields satisfy the full interface equations. Let $C_P(D^{\ast })$ satisfy $\mathrm{\parallel }v{\mathrm{\parallel }}_{L^2(D^{\ast })}\le C_P(D^{\ast })\mathrm{\parallel }v{\mathrm{\parallel }}_V$ for $v\in V$. Impose the time-step restriction ${\tau }_{\max }\mathrm{\Lambda }\le \frac{1}{2}$; no other mesh/time coupling is imposed.

At this Sobolev regularity, explicitly adopt the following distributional extension of the physical momentum balance: for almost every $t^{\ast }\in (0,T^{\ast })$, every $v\in V$, and every $q\in Q$. Here $\epsilon (w)=(\mathrm{\nabla }w+\mathrm{\nabla }{w}^{\mathsf{T}})\mathrm{/}2$.

$({\mathrm{\partial }}_{t^{\ast }}{u}^{\ast },v{)}_{L^2(D^{\ast })}+2{\nu }^{\ast }{\int }_{D^{\ast }}\epsilon (u^{\ast }):\epsilon (v)dy+{\int }_{D^{\ast }}(u^{\ast }\cdot \mathrm{\nabla })u^{\ast }\cdot vdy-{\int }_{D^{\ast }}{p}^{\ast }\mathrm{div}vdy=(f^{\ast },v{)}_{L^2(D^{\ast })},-{\int }_{D^{\ast }}q\mathrm{div}{u}^{\ast }dy=0.$

The physical weak stress pairing uses $\phi (x):=v(x\mathrm{/}{\mathrm{\ell }}_0)\in H_0^1({\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}};{\mathbb{R}}^3)$ and is $2{\mu }_0{\int }_{{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}{D}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}:{\epsilon }_x(\phi )dx$, where ${\epsilon }_x(\phi )=({\mathrm{\nabla }}_x\phi +{\mathrm{\nabla }}_x{\phi }^{\mathsf{T}})\mathrm{/}2$; the declared scaling gives the displayed $2{\nu }^{\ast }$ strain term. The MINI interface's real skew convective form is $c(a;v,w)=\frac{1}{2}{\int }_{D^{\ast }}((a\cdot \mathrm{\nabla })v\cdot w-(a\cdot \mathrm{\nabla })w\cdot v)dy$; for the divergence-free, zero-trace $u^{\ast }$ above, integration by parts gives $c(u^{\ast };u^{\ast },v)={\int }_{D^{\ast }}(u^{\ast }\cdot \mathrm{\nabla })u^{\ast }\cdot v$. Thus the displayed weak equation is exactly the continuous exact-field equation in the MINI interface. This weak formulation is explicitly adopted at the stated Sobolev regularity; those hypotheses do not imply the physical model definition's classical $C^2\mathrm{/}{C}^1$ regularity. It asserts no extension to weak limits or lower-regularity solutions.

Set $M=1+\mathrm{\parallel }{u}^{\ast }{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}{H}^2}+\mathrm{\parallel }{p}^{\ast }{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}{H}^1},H=hM+{\tau }_{\max }{M}^2,M_u=\mathrm{\parallel }{u}^{\ast }{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(0,T^{\ast };V)},\mathrm{\Lambda }=1+C_D({\nu }^{\ast }{)}^{-3}{M}_u^4,$ $M_{\mathrm{\infty }}=(\mathrm{\parallel }{u}^{\ast }(0){\mathrm{\parallel }}_2^2+C_P(D^{\ast }{)}^2({\nu }^{\ast }{)}^{-1}\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_{L^2{L}^2}^2{)}^{1\mathrm{/}2},M_E=({\nu }^{\ast }{)}^{-1\mathrm{/}2}\mathrm{\parallel }{u}^{\ast }(0){\mathrm{\parallel }}_2+C_P(D^{\ast })({\nu }^{\ast }{)}^{-1}\mathrm{\parallel }{f}^{\ast }{\mathrm{\parallel }}_{L^2{L}^2},$ $C_{\ast }=C_{D,\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}}(1+{\nu }^{\ast }{)}^2(1+({\nu }^{\ast }{)}^{-4})(1+T^{\ast }{)}^2(1+M_{\mathrm{\infty }}+M_E+M{)}^4\exp (2T^{\ast }\mathrm{\Lambda })\mathrm{.}$ Here $C_D$ depends only on the domain, dimension, and trilinear-form constant, and $C_{D,\mathrm{s}\mathrm{h}\mathrm{a}\mathrm{p}\mathrm{e}}$ is independent of $h$, the time steps, and ${\nu }^{\ast }$. For the pressure estimate let $\beta >0$ be the uniform MINI inf-sup lower bound, $M_b=\sqrt{3}$, let $C_I$ satisfy ${\inf }_{q_h\in Q_h}\mathrm{\parallel }q-q_h{\mathrm{\parallel }}_2\le C_{I}h\mathrm{\parallel }q{\mathrm{\parallel }}_{H^1(D^{\ast })}$ for $q\in H^1(D^{\ast })\cap Q$, and choose $C_c$ with $\mathrm{\mid }c(a;b,v)\mathrm{\mid }\le C_c\mathrm{\parallel }a{\mathrm{\parallel }}_V\mathrm{\parallel }b{\mathrm{\parallel }}_V\mathrm{\parallel }v{\mathrm{\parallel }}_V$. Define $K=(1+M_b\mathrm{/}\beta )C_I{T}^{\ast }+{\beta }^{-1}(2C_P(D^{\ast })+[2{\nu }^{\ast }\sqrt{T^{\ast }}+C_c(M_E+\sqrt{T^{\ast }}{M}_u)]({\nu }^{\ast }{)}^{-1\mathrm{/}2})C_{\ast }+{\beta }^{-1}(2{\nu }^{\ast }{T}^{\ast }+2C_c{T}^{\ast })\mathrm{.}$ For bounded endpoint velocity sensors ${\mathrm{\ell }}_n:V\to \mathbb{R}$ that are real-linear and continuous, define $\mathrm{\parallel }{\mathrm{\ell }}_n{\mathrm{\parallel }}_{V^{\ast }}={\sup }_{\mathrm{\parallel }v{\mathrm{\parallel }}_V\le 1}\mathrm{\mid }{\mathrm{\ell }}_n(v)\mathrm{\mid }$, $\mathcal{O}(\mathrm{\ell };u^{\ast })={\sum }_{n=1}^N{\tau }_n{\mathrm{\ell }}_n(u^{\ast }(t_n))$, ${\mathcal{O}}_h(\mathrm{\ell };u_h^{\ast })={\sum }_{n=1}^N{\tau }_n{\mathrm{\ell }}_n(u_h^{\ast ,n})$, and $\mathrm{\parallel }\mathrm{\ell }{\mathrm{\parallel }}_{{\mathrm{\ell }}_{\tau }^2(V^{\ast })}=({\sum }_{n=1}^N{\tau }_n\mathrm{\parallel }{\mathrm{\ell }}_n{\mathrm{\parallel }}_{V^{\ast }}^2{)}^{1\mathrm{/}2}$. The dimensionless interface observations obey $\mathrm{\mid }{\mathcal{O}}_h(\mathrm{\ell };u_h)-\mathcal{O}(\mathrm{\ell };u^{\ast })\mathrm{\mid }\le ({\nu }^{\ast }{)}^{-1\mathrm{/}2}{C}_{\ast }H{\left({\sum }_{n=1}^N{\tau }_n\mathrm{\parallel }{\mathrm{\ell }}_n{\mathrm{\parallel }}_{V^{\ast }}^2\right)}^{1\mathrm{/}2}\mathrm{.}$ For every $\xi \in Q$, every $\eta \in C_c^{\mathrm{\infty }}(0,T^{\ast })$, and every node $0\le m\le N$, with $P^{\ast }(t)={\int }_0^t{p}^{\ast }(s)ds$, $P_h^{\ast ,m}={\sum }_{n=1}^m{\tau }_n{p}_h^{\ast ,n}$, and ${\bar{p}}_h^{\ast }(t)=p_h^{\ast ,n}$ on $(t_{n-1},t_n]$, the dimensionless pressure observations obey $\mathrm{\mid }(P^{\ast }(t_m)-P_h^{\ast ,m},\xi {)}_{L^2}\mathrm{\mid }\le KH\mathrm{\parallel }\xi {\mathrm{\parallel }}_Q,\mid {({\int }_0^{T^{\ast }}(p^{\ast }-{\bar{p}}_h^{\ast })\eta d{t}^{\ast },\xi )}_{L^2}\mid \le (K+1)H\mathrm{\parallel }{\eta }^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^1(0,T^{\ast })}\mathrm{\parallel }\xi {\mathrm{\parallel }}_Q\mathrm{.}$ The physical endpoint velocity reading obtained by multiplying each normalized scalar sensor by $U_0$ has error bound $U_0({\nu }^{\ast }{)}^{-1\mathrm{/}2}{C}_{\ast }H\mathrm{\parallel }\mathrm{\ell }{\mathrm{\parallel }}_{{\mathrm{\ell }}_{\tau }^2(V^{\ast })}$. Define the physical primitive $P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,t)={\int }_0^t{p}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,s)ds$, the physical discrete primitive $P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},h}^m(x)={\sum }_{n=1}^m({\mathrm{\ell }}_0\mathrm{/}{U}_0){\tau }_n{\rho }_0{U}_0^2{p}_h^{\ast ,n}(x\mathrm{/}{\mathrm{\ell }}_0)$, and the physical right-constant pressure reconstruction ${\bar{p}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},h}(x,t)={\rho }_0{U}_0^2{\bar{p}}_h^{\ast }(x\mathrm{/}{\mathrm{\ell }}_0,U_{0}t\mathrm{/}{\mathrm{\ell }}_0)$. For the physical node $t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},m}=({\mathrm{\ell }}_0\mathrm{/}{U}_0)t_m$, where $t_m$ is the dimensionless partition node, the spatially weighted physical primitive error ${\int }_{{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\xi (x\mathrm{/}{\mathrm{\ell }}_0)(P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,t_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},m})-P_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},h}^m(x))dx$ is bounded by ${\rho }_0{U}_0{\mathrm{\ell }}_0^{4}KH\mathrm{\parallel }\xi {\mathrm{\parallel }}_Q$. The physical volume-time pressure error $\mid {\int }_0^{({\mathrm{\ell }}_0\mathrm{/}{U}_0)T^{\ast }}{\int }_{{\mathrm{\Omega }}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}}\xi (x\mathrm{/}{\mathrm{\ell }}_0)(p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}(x,t)-{\bar{p}}_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s},h}(x,t))\eta (U_{0}t\mathrm{/}{\mathrm{\ell }}_0)dxdt\mid$ is at most ${\rho }_0{U}_0{\mathrm{\ell }}_0^4(K+1)H\mathrm{\parallel }{\eta }^{\mathrm{\prime }}{\mathrm{\parallel }}_{L^1(0,T^{\ast })}\mathrm{\parallel }\xi {\mathrm{\parallel }}_Q$ in $\mathrm{P}\mathrm{a}{\mathrm{m}}^3\mathrm{s}$. The scale factor follows from $p_{\mathrm{p}\mathrm{h}\mathrm{y}\mathrm{s}}={\rho }_0{U}_0^2{p}^{\ast }$, $dx={\mathrm{\ell }}_0^{3}dy$, and $dt=({\mathrm{\ell }}_0\mathrm{/}{U}_0)d{t}^{\ast }$. These are conditional model predictions on the strong-solution branch. They make no Reynolds-uniform claim and no instantaneous pressure-error claim, and do not establish solver convergence or empirical validation.

Forced constant-density incompressible Newtonian branch

Unforced constant-density Newtonian model conventions

Selected stationary no-slip boundary data

Smooth MINI observation interface

Bounded velocity-observation error for the smooth MINI branch

Accumulated pressure-observation error for the smooth MINI branch

Tong, Fluid Mechanics, §§3.1.1 and 3.2.1

Powers, Lecture Notes on Intermediate Fluid Mechanics, §5.4.1

Guermond, MATH 661, Chapter 54, §54.2

Boffi–Brezzi–Fortin, Mixed Finite Element Methods and Applications, §§8.4.2 and 8.6.1